MTH120 College Algebra Chapter 5.3

 5.3 Graphs of Polynomial Functions:

Graphs of polynomial functions exhibit various characteristics and behaviors based on the degree and leading coefficient of the polynomial. Here are some key points about the graphs of polynomial functions:

1. Degree of the Polynomial:

   • The degree of a polynomial function is determined by the highest power of the variable (usually $�$) in the polynomial expression.
   • A polynomial of degree $�$ can have at most $�-1$ turning points (local extrema) on its graph.

2. Leading Coefficient:

   • The leading coefficient is the coefficient of the term with the highest power of the variable.
   • It determines the overall behavior of the graph at the far ends as $�$ approaches $-\mathrm{\infty }$ and $+\mathrm{\infty }$.

3. End Behavior:

   • The end behavior of a polynomial function depends on its degree and leading coefficient:
     • For even-degree polynomials:
       • If the leading coefficient is positive, the graph rises on both sides as $�$ goes to $-\mathrm{\infty }$ and $+\mathrm{\infty }$.
       • If the leading coefficient is negative, the graph falls on both sides as $�$ goes to $-\mathrm{\infty }$ and $+\mathrm{\infty }$.
     • For odd-degree polynomials:
       • If the leading coefficient is positive, the graph rises to the right ($+\mathrm{\infty }$) and falls to the left ($-\mathrm{\infty }$).
       • If the leading coefficient is negative, the graph falls to the right ($+\mathrm{\infty }$) and rises to the left ($-\mathrm{\infty }$).

4. Zeros and X-Intercepts:

   • Zeros or x-intercepts of a polynomial function are the values of $�$ for which $�(�)=0$.
   • The number of real zeros of a polynomial is equal to its degree.
   • The graph crosses the x-axis at x-intercepts.

5. Y-Intercept:

   • The y-intercept is the point where the graph intersects the y-axis.
   • To find it, evaluate the function at $�=0$, which gives you the value $�(0)$. The point $(0,�(0))$ is the y-intercept.

6. Turning Points (Local Extrema):

   • Turning points are points on the graph where the function changes direction.
   • The number of turning points is at most one less than the degree of the polynomial.

7. Ends of the Graph:

   • The ends of the graph may have different behavior depending on whether the degree is even or odd and whether the leading coefficient is positive or negative.

8. Multiplicity of Zeros:

   • When a zero has multiplicity greater than 1, the graph touches or bounces off the x-axis at that point without crossing it.

9. Continuous:

   • Polynomial functions are continuous over their entire domain.

In summary, the graph of a polynomial function exhibits various behaviors based on its degree, leading coefficient, and the presence of multiple roots. Understanding these characteristics helps in sketching and analyzing polynomial functions and their graphs.

Recognizing the characteristics of graphs of polynomial functions is essential for understanding and analyzing their behavior. Here are some key characteristics to look for when dealing with the graphs of polynomial functions:

1. Continuity:

   • Polynomial functions are continuous over their entire domain.

Recognizing these characteristics and analyzing the graph of a polynomial function can provide valuable information about its properties, such as the presence of roots, turning points, and its behavior at extreme values of $�$.

Factoring is a powerful method for finding the zeros (roots) of polynomial functions. To find the zeros of a polynomial function using factoring, follow these steps:

1. Write the Polynomial Equation:

   • Express the polynomial function as an equation by setting it equal to zero. For example, if you have the polynomial $�(�)=2{�}^2-5�-3$, you would write the equation $2{�}^2-5�-3=0$.

2. Factor the Polynomial:

   • Attempt to factor the polynomial on the left side of the equation. Factoring involves expressing the polynomial as a product of simpler polynomial expressions.
   • For example, in the equation $2{�}^2-5�-3=0$, you can factor it as $(2�+1)(�-3)=0$ by finding two binomials that multiply to the original polynomial.

3. Set Each Factor Equal to Zero:

   • Once you have factored the polynomial, set each factor equal to zero and solve for $�$.
   • In the example above, you would set $2�+1=0$ and $�-3=0$ and solve for $�$:
     • $2�+1=0$ gives $�=-\frac{1}{2}$.
     • $�-3=0$ gives $�=3$.

4. Find the Zeros:

   • The solutions you found in step 3 are the zeros (roots) of the polynomial function. In this case, the zeros are $�=-\frac{1}{2}$ and $�=3$.

5. Verify the Zeros:

   • Verify your solutions by substituting them back into the original polynomial equation to ensure they make it true. If the equation holds true for the values you found, they are indeed the zeros.

6. Express the Zeros:

   • Express the zeros as ordered pairs $(�,0)$ if you want to represent them graphically.

In summary, factoring allows you to break down a polynomial into its factors, which makes it easier to find the values of $�$ that make the polynomial equal to zero. These values are the zeros of the polynomial, and they are important for understanding the behavior of the function and graphing it.

Identifying zeros (roots) and their multiplicities is crucial when working with polynomial functions. The multiplicity of a zero indicates how many times a specific root appears in the factored form of the polynomial. Here's how to identify zeros and their multiplicities:

1. Write the Polynomial Equation:

   • Start with the polynomial equation. For example, consider the polynomial $�(�)=(�-2{)}^3(�+1)(�-3{)}^2$.

2. Factor the Polynomial:

   • Factor the polynomial completely. In the example, the polynomial is already factored.

3. Identify the Zeros:

   • Zeros are values of $�$ that make the polynomial equal to zero. To find zeros, set each factor equal to zero and solve for $�$.
   • In the example, set each factor equal to zero:
     • $(�-2{)}^3=0$: This has a zero of 2 with a multiplicity of 3.
     • $�+1=0$: This has a zero of -1 with a multiplicity of 1.
     • $(�-3{)}^2=0$: This has a zero of 3 with a multiplicity of 2.

4. Count the Multiplicity:

   • The multiplicity of a zero is the number of times it appears as a factor with a corresponding power in the factored form of the polynomial.
   • In the example, the multiplicities are:
     • Zero 2 has a multiplicity of 3 (appears as a factor three times).
     • Zero -1 has a multiplicity of 1 (appears as a factor once).
     • Zero 3 has a multiplicity of 2 (appears as a factor twice).

5. Express the Zeros with Multiplicity:

   • Write the zeros along with their multiplicities as ordered pairs $(�,\text{multiplicity})$ if you want to represent them with multiplicity information.

In summary, when identifying zeros and their multiplicities:

• Find the values of $�$ that make the polynomial equal to zero by setting each factor equal to zero.
• Count the number of times each zero appears as a factor with a corresponding power in the factored form of the polynomial to determine its multiplicity.
• Express the zeros with their multiplicities to provide a complete description of the polynomial's roots. Multiplicity affects the behavior of the graph near each zero.

The graphical behavior of polynomial functions at their x-intercepts (zeros or roots) depends on the multiplicity of each x-intercept. The multiplicity of an x-intercept indicates how many times that root appears as a factor in the factored form of the polynomial. Here are the common scenarios:

1. Simple Zero (Multiplicity = 1):

   • If an x-intercept has multiplicity 1, it means that it appears as a single linear factor in the factored form of the polynomial.
   • The graph of the polynomial crosses the x-axis at this point, just as a linear function does.
   • For example, consider the polynomial $�(�)=(�-2)(�+3)$. The zeros are $�=2$ and $�=-3$, both with multiplicity 1.

   

2. Double Zero (Multiplicity = 2):

   • If an x-intercept has multiplicity 2, it means that it appears as a quadratic factor in the factored form of the polynomial.
   • The graph touches the x-axis at this point but does not cross it (similar to the behavior of a parabola at its vertex).
   • For example, consider the polynomial $�(�)=(�-2{)}^2(�+3)$. The zero $�=2$ has multiplicity 2, and $�=-3$ has multiplicity 1.

   

3. Triple Zero (Multiplicity = 3):

   • If an x-intercept has multiplicity 3, it means that it appears as a cubic factor in the factored form of the polynomial.
   • The graph behaves like a cubic function at this point, crossing the x-axis and changing direction.
   • For example, consider the polynomial $�(�)=(�-2{)}^3(�+3)$. The zero $�=2$ has multiplicity 3, and $�=-3$ has multiplicity 1.

   

4. Higher Multiplicity:

   • The behavior becomes more pronounced as the multiplicity increases. For even higher multiplicities (e.g., 4, 5, etc.), the graph flattens out even more at the x-intercept before changing direction.

   

In summary, the graphical behavior of a polynomial function at its x-intercepts depends on the multiplicity of each root. Simple zeros lead to crossing the x-axis, double zeros create points of tangency, and higher multiplicities result in flatter points of contact before changing direction. Understanding the multiplicity of x-intercepts helps in sketching accurate graphs of polynomial functions.

Determining the end behavior of a polynomial function involves understanding how the function behaves as the independent variable (usually $�$) approaches positive infinity ($+\mathrm{\infty }$) and negative infinity ($-\mathrm{\infty }$). The end behavior is determined by the degree and leading coefficient of the polynomial. Here are the key principles for determining end behavior:

1. Degree of the Polynomial:

   • The degree of a polynomial function is the highest power of $�$ in the polynomial expression. It determines the overall shape of the graph.

2. Leading Coefficient:

   • The leading coefficient is the coefficient of the term with the highest power of $�$.
   • It plays a critical role in determining whether the end behavior rises or falls.

3. Odd-Degree Polynomials:

   • For odd-degree polynomials (e.g., ${�}^3$, ${�}^5$), the end behavior is as follows:
     • If the leading coefficient is positive, the graph rises to the right ($�$ approaches $+\mathrm{\infty }$) and falls to the left ($�$ approaches $-\mathrm{\infty }$).
     • If the leading coefficient is negative, the graph falls to the right ($�$ approaches $+\mathrm{\infty }$) and rises to the left ($�$ approaches $-\mathrm{\infty }$).

4. Even-Degree Polynomials:

   • For even-degree polynomials (e.g., ${�}^2$, ${�}^4$), the end behavior is as follows:
     • If the leading coefficient is positive, the graph rises on both sides ($�$ approaches $+\mathrm{\infty }$ and $-\mathrm{\infty }$) or falls on both sides.
     • If the leading coefficient is negative, the graph falls on both sides ($�$ approaches $+\mathrm{\infty }$ and $-\mathrm{\infty }$) or rises on both sides.

5. Example 1:

   • For the polynomial $�(�)=3{�}^3-2{�}^2+5�-1$:
     • The degree is 3 (odd), and the leading coefficient is 3 (positive).
     • As $�$ approaches $+\mathrm{\infty }$, the graph rises to the right, and as $�$ approaches $-\mathrm{\infty }$, the graph falls to the left.

6. Example 2:

   • For the polynomial $�(�)=-2{�}^4+6{�}^3-4{�}^2+7$:
     • The degree is 4 (even), and the leading coefficient is -2 (negative).
     • As $�$ approaches $+\mathrm{\infty }$ and $-\mathrm{\infty }$, the graph falls on both sides.

To summarize, the end behavior of a polynomial function is determined by its degree and leading coefficient. Understanding this behavior helps you sketch the graph of the polynomial and provides insights into how the function behaves at extreme values of $�$.

Understanding the relationship between the degree of a polynomial and the number of turning points (local extrema) on its graph is essential in polynomial function analysis. The degree of a polynomial is directly related to the maximum number of turning points it can have. Here's how the two are connected:

1. Degree of a Polynomial:

   • The degree of a polynomial is determined by the highest power of the variable (usually $�$) in the polynomial expression.
   • For example, the degree of $�(�)=2{�}^3-3{�}^2+4�-1$ is 3 because the highest power of $�$ is 3.

2. Number of Turning Points:

   • A turning point (or local extremum) is a point on the graph where the function changes direction, either from increasing to decreasing or from decreasing to increasing.
   • The maximum number of turning points a polynomial of degree $�$ can have is $�-1$.

Now, let's break down the relationship based on the degree of the polynomial:

• Degree 0 (Constant Function):

  • A degree-0 polynomial is a constant function, e.g., $�(�)=5$.
  • It has no turning points; the graph is a horizontal line.

• Degree 1 (Linear Function):

  • A degree-1 polynomial is a linear function, e.g., $�(�)=3�+2$.
  • It has at most 0 turning points; the graph is a straight line.

• Degree 2 (Quadratic Function):

  • A degree-2 polynomial is a quadratic function, e.g., $�(�)={�}^2-4$.
  • It can have at most 1 turning point; the graph is a parabola.

• Degree 3 (Cubic Function):

  • A degree-3 polynomial is a cubic function, e.g., $�(�)={�}^3+2{�}^2-�-3$.
  • It can have at most 2 turning points.

• Degree 4 (Quartic Function):

  • A degree-4 polynomial is a quartic function, e.g., $�(�)={�}^4-4{�}^3+6{�}^2-4�+1$.
  • It can have at most 3 turning points.

• Degree 5 (Quintic Function):

  • A degree-5 polynomial is a quintic function, e.g., $�(�)={�}^5+2{�}^4-3{�}^3+4{�}^2-5�+1$.
  • It can have at most 4 turning points.

And so on. As the degree of the polynomial increases, the maximum number of turning points also increases by one. However, not all polynomials of a given degree will necessarily have the maximum number of turning points. The actual number of turning points can be fewer than the maximum, depending on the specific coefficients and factors of the polynomial.

In summary, the degree of a polynomial determines the maximum number of turning points that can exist on its graph. Understanding this relationship helps in analyzing and visualizing the behavior of polynomial functions.

Graphing polynomial functions involves several steps to understand and visualize the behavior of the function. Here's a step-by-step guide on how to graph polynomial functions:

Step 1: Understand the Polynomial Function

• Begin by examining the polynomial function and identifying its key characteristics:
  • Degree: Determine the highest power of the variable (usually $�$) in the polynomial.
  • Leading Coefficient: Identify the coefficient of the term with the highest power.
  • Zeros and Multiplicities: Find the zeros (roots) and their multiplicities by setting the polynomial equal to zero and factoring it.

Step 2: Determine the End Behavior

• Use the degree and leading coefficient to determine the end behavior of the graph as $�$ approaches positive infinity and negative infinity:
  • For odd-degree polynomials:
    • If the leading coefficient is positive, the graph rises to the right and falls to the left.
    • If the leading coefficient is negative, the graph falls to the right and rises to the left.
  • For even-degree polynomials:
    • If the leading coefficient is positive, the graph rises on both sides or falls on both sides.
    • If the leading coefficient is negative, the graph falls on both sides or rises on both sides.

Step 3: Find and Plot the Zeros

• Plot the x-intercepts (zeros) on the graph. The zeros are the values of $�$ where the polynomial equals zero.
• Use the multiplicities of the zeros to determine the behavior at each zero:
  • For simple zeros (multiplicity = 1), the graph crosses the x-axis.
  • For double zeros (multiplicity = 2), the graph touches the x-axis but doesn't cross it.
  • For higher multiplicities, the graph exhibits more complex behavior.

Step 4: Identify and Plot Turning Points

• Determine the number of turning points (local extrema) based on the degree of the polynomial:
  • A degree-$�$ polynomial can have at most $�-1$ turning points.
• Locate the turning points by finding the critical points (where the derivative is zero) and analyzing their nature (minima, maxima, or inflection points).
• Plot these turning points on the graph.

Step 5: Sketch the Graph

• Based on the information gathered in the previous steps, sketch the graph of the polynomial function. Pay attention to the end behavior, zeros, and turning points.
• Use smooth curves to connect the various parts of the graph.

Step 6: Label Key Features

• Label key points on the graph, such as the x-intercepts, turning points, and any other relevant information.
• Include axis labels and a title for clarity.

Step 7: Check Your Work

• Verify that your graph accurately represents the polynomial function by confirming that it passes through the correct x-intercepts, has the expected end behavior, and exhibits the right number of turning points.

Step 8: Use Technology (Optional)

• You can use graphing calculators or computer software to assist in graphing more complex polynomial functions.

Graphing polynomial functions can be a challenging task for higher-degree polynomials, but following these steps systematically will help you visualize and understand their behavior on the coordinate plane.

The Intermediate Value Theorem (IVT) is a fundamental theorem in calculus that helps establish the existence of roots (zeros) for continuous functions. It states that if a continuous function $�(�)$ takes on two different signs at two points $�$ and $�$ within an interval $[�,�]$, then there exists at least one point $�$ in the interval $[�,�]$ where $�(�)=0$.

Here's how to use the Intermediate Value Theorem to find approximate solutions (roots) for a continuous function within a given interval:

1. Understand the Theorem:

   • Ensure that the function $�(�)$ is continuous on the closed interval $[�,�]$.
   • Verify that $�(�)$ and $�(�)$ have different signs. This means that $�(�)\cdot �(�)<0$.

2. Choose an Interval:

   • Select a closed interval $[�,�]$ that contains the suspected root (zero) or where you want to find the root.
   • Make sure the function is continuous within this interval.

3. Evaluate $�(�)$ and $�(�)$:

   • Compute the values of $�(�)$ and $�(�)$ by plugging in the endpoints $�$ and $�$ into the function $�(�)$.

4. Check for a Sign Change:

   • Determine whether $�(�)$ and $�(�)$ have different signs. If $�(�)\cdot �(�)<0$, this indicates a sign change, which is a requirement for IVT.

5. Apply IVT:

   • Since $�(�)$ and $�(�)$ have different signs, IVT guarantees the existence of at least one point $�$ in the interval $[�,�]$ where $�(�)=0$.
   • In other words, there is a root (zero) of the function $�(�)$ within the interval $[�,�]$.

6. Narrow Down the Root (Optional):

   • You can use numerical methods like the bisection method, Newton's method, or a calculator to refine the estimate of the root within the interval $[�,�]$.

Here's an example to illustrate how to use the Intermediate Value Theorem:

Example: Consider the function $�(�)={�}^3-2{�}^2+�-1$. Determine if there is a root of this function in the interval $[1,2]$.

1. Check that $�(�)$ is continuous on $[1,2]$ (it is a polynomial, so it is continuous everywhere).
2. Choose the interval $[1,2]$.
3. Evaluate $�(1)=-1$ and $�(2)=1$.
4. Since $�(1)\cdot �(2)=(-1)\cdot 1<0$, there is a sign change.
5. Apply the Intermediate Value Theorem to conclude that there exists at least one root of $�(�)$ in the interval $[1,2]$.

In this example, the Intermediate Value Theorem doesn't tell us the exact value of the root, but it guarantees that there is at least one root within the specified interval. Numerical methods can be used to approximate the root further if needed.

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus and real analysis that relates to the behavior of continuous functions on intervals. The theorem states that if a real-valued function $�(�)$ is continuous on a closed interval $[�,�]$ and $�$ is any number between $�(�)$ and $�(�)$ (where $�(�)\mathrm{\ne }�(�)$), then there exists at least one number $�$ in the open interval $(�,�)$ such that $�(�)=�$.

In simpler terms, the IVT guarantees that if a continuous function takes on two distinct values $�(�)$ and $�(�)$ at the endpoints of a closed interval, then the function must also take on every value between $�(�)$ and $�(�)$ somewhere within the interval. In other words, the graph of the function must cross every horizontal line segment between $�(�)$ and $�(�)$ at least once.

Key points and implications of the Intermediate Value Theorem:

1. Continuity Requirement: The function $�(�)$ must be continuous on the entire closed interval $[�,�]$.

2. Closed Interval: The interval $[�,�]$ must be a closed interval, meaning it includes its endpoints $�$ and $�$.

3. Distinct Values: The values $�(�)$ and $�(�)$ must be distinct, i.e., $�(�)\mathrm{\ne }�(�)$.

4. Existence of a Root: The IVT implies the existence of at least one $�$ in the open interval $(�,�)$ where $�(�)=�$.

5. Multiple Roots: There can be more than one $�$ in the interval $(�,�)$ where $�(�)=�$, depending on the behavior of the function.

6. Applications: The IVT is often used to prove the existence of solutions (roots) for equations, inequalities, and various real-world problems. It's a fundamental tool in calculus and real analysis.

The Intermediate Value Theorem provides an important connection between the continuity of a function and the existence of solutions, and it's used extensively in calculus and mathematical analysis to establish important results and solve problems.

Writing formulas for polynomial functions involves expressing a polynomial as a function of a variable (typically $�$) using its coefficients and powers of $�$. The general form of a polynomial function is:

$�(�)={�}_{�}{�}^{�}+{�}_{�-1}{�}^{�-1}+\dots +{�}_2{�}^2+{�}_1�+{�}_0$

In this formula:

• $�(�)$ represents the function.
• ${�}_{�},{�}_{�-1},\dots ,{�}_2,{�}_1,{�}_0$ are the coefficients of the polynomial. ${�}_{�}$ is the leading coefficient, ${�}_0$ is the constant term, and the other coefficients correspond to the terms with powers of $�$ from highest to lowest.
• $�$ is the variable.
• $�$ is the degree of the polynomial, which is the highest power of $�$ in the polynomial.

To write the formula for a specific polynomial function, follow these steps:

1. Determine the Degree and Coefficients:

   • Identify the degree of the polynomial by finding the highest power of $�$ that appears.
   • Identify the coefficients of each term in the polynomial.

2. Write the Function:

   • Write the polynomial function using the general form, replacing ${�}_{�},{�}_{�-1},\dots ,{�}_2,{�}_1,{�}_0$ with the actual coefficients you identified.
   • Include the variable $�$.

3. Example: Let's say you have the polynomial $�(�)=3{�}^3-2{�}^2+5�-1$.

   • The degree of the polynomial is 3.
   • The coefficients are: ${�}_3=3$, ${�}_2=-2$, ${�}_1=5$, and ${�}_0=-1$.
   • Using the general form, the formula for this polynomial is: $�(�)=3{�}^3-2{�}^2+5�-1$

4. Simplify if Necessary:

   • Simplify the polynomial function if there are any like terms that can be combined.

5. Final Formula:

   • Write the final, simplified formula for the polynomial function.

That's how you write formulas for polynomial functions. The key is to identify the degree and coefficients and then use the general form to express the polynomial as a function of $�$.

The factored form of a polynomial expresses the polynomial as a product of its linear factors. This form is useful for finding the roots (zeros) of the polynomial and understanding its behavior. To write a polynomial in factored form, follow these steps:

1. Identify the Zeros (Roots): Find the values of $�$ for which the polynomial equals zero. These are the zeros (roots) of the polynomial. You can do this by setting the polynomial equal to zero and solving for $�$.

2. Write Linear Factors: For each zero $�$, write a linear factor $(�-�)$ in which $�$ is the root.

3. Combine Factors: Multiply all the linear factors together to obtain the factored form of the polynomial.

Here's a step-by-step example:

Example: Factor the polynomial $�(�)={�}^3-4{�}^2-11�+30$.

Step 1: Identify the Zeros (Roots): To find the roots, set $�(�)$ equal to zero and solve for $�$: ${�}^3-4{�}^2-11�+30=0$

You can use various methods like the rational root theorem, synthetic division, or a graphing calculator to find the roots. In this case, you find that the roots are $�=-3$, $�=2$, and $�=5$.

Step 2: Write Linear Factors: For each root, write a linear factor:

For $�=-3$, the linear factor is $(�-(-3))=(�+3)$.

For $�=2$, the linear factor is $(�-2)$.

For $�=5$, the linear factor is $(�-5)$.

Step 3: Combine Factors: Multiply the linear factors together to obtain the factored form: $�(�)=(�+3)(�-2)(�-5)$

So, the factored form of the polynomial $�(�)$ is $(�+3)(�-2)(�-5)$.

You can verify this by multiplying the factors back together to see that it equals the original polynomial. Factoring polynomials can be more complex for higher-degree polynomials, but the process remains the same: identify the zeros and write them as linear factors.

Local and global extrema are important concepts in calculus and optimization. They refer to the maximum and minimum values of a function within specific intervals or over its entire domain, respectively. Understanding these extrema helps analyze functions and solve optimization problems. Here's how to use local and global extrema:

Local Extrema:

Local extrema are the maximum and minimum values of a function within a specific interval. To find local extrema, follow these steps:

1. Identify Critical Points: Critical points are values of $�$ where the derivative of the function is zero or undefined. Set ${�}^{\mathrm{\prime }}(�)=0$ and solve for $�$ to find these points.

2. Use the First Derivative Test: Evaluate the sign of the derivative ${�}^{\mathrm{\prime }}(�)$ on both sides of each critical point.

   • If ${�}^{\mathrm{\prime }}(�)$ changes from positive to negative as you move from the left to the right of a critical point, there is a local maximum at that point.
   • If ${�}^{\mathrm{\prime }}(�)$ changes from negative to positive as you move from the left to the right of a critical point, there is a local minimum at that point.

3. Check Endpoints (if applicable): If the interval includes endpoints (e.g., $[�,�]$), evaluate the function at the endpoints to check for potential extrema.

Global Extrema:

Global extrema are the maximum and minimum values of a function over its entire domain. To find global extrema, follow these steps:

1. Identify Critical Points: As in the case of local extrema, find the critical points by setting ${�}^{\mathrm{\prime }}(�)=0$ and solving for $�$.

2. Use the First Derivative Test: Determine whether each critical point corresponds to a local maximum, minimum, or neither, as explained above.

3. Check the Function's Values at Endpoints: If the function is defined on a closed interval $[�,�]$, evaluate it at the endpoints $�$ and $�$.

4. Compare Values: Compare the values obtained in step 2 and step 3.

   • The largest value among the local maxima and the endpoint values is the global maximum.
   • The smallest value among the local minima and the endpoint values is the global minimum.

Example:

Consider the function $�(�)={�}^3-3{�}^2-4�+5$ on the interval $[-1,4]$.

1. Critical Points: Find the critical points by solving ${�}^{\mathrm{\prime }}(�)=0$. You find $�=-1$, $�=2$, and $�=3$ as critical points.

2. First Derivative Test: Apply the First Derivative Test to the critical points:

   • At $�=-1$, ${�}^{\mathrm{\prime }}(-1)<0$, so there is a local maximum.
   • At $�=2$, ${�}^{\mathrm{\prime }}(2)>0$, so there is a local minimum.
   • At $�=3$, ${�}^{\mathrm{\prime }}(3)<0$, so there is a local maximum.

3. Endpoint Values: Evaluate $�(-1)$ and $�(4)$ to check the function's values at the endpoints. You find $�(-1)=13$ and $�(4)=21$.

4. Global Extrema:

   • The global maximum is 21 (at $�=4$).
   • The global minimum is 13 (at $�=-1$).

In this example, you've determined both the local and global extrema of the function. Local extrema are within specific intervals (between critical points), while global extrema are over the entire interval $[-1,4]$.

Do all polynomial functions have a global minimum or maximum?

Not all polynomial functions have a global minimum or maximum. Whether a polynomial function has a global minimum or maximum depends on the degree and characteristics of the polynomial as well as the interval over which it is defined.

Here are some key points to consider:

1. Polynomials of Odd Degree:

   • Polynomials of odd degree (e.g., $�(�)=�{�}^3+�{�}^2+��+�$) do not have a global minimum or maximum over their entire domain. They may have local extrema (local minimum or maximum) within specific intervals, but they extend infinitely in both directions without reaching a global extreme value.

2. Polynomials of Even Degree:

   • Polynomials of even degree (e.g., $�(�)=�{�}^4+�{�}^3+�{�}^2+��+�$) may or may not have a global minimum or maximum.
   • If the leading coefficient (the coefficient of the highest power of $�$) is positive, and the degree is even, the polynomial tends to positive infinity as $�$ approaches both positive and negative infinity. In this case, it has neither a global minimum nor a global maximum.
   • If the leading coefficient is negative, and the degree is even, the polynomial tends to negative infinity as $�$ approaches both positive and negative infinity. Again, it has neither a global minimum nor a global maximum.

3. Bounded Polynomials:

   • Some polynomial functions are bounded by horizontal lines, meaning they do have global minimum and maximum values.
   • For example, a quadratic polynomial with a positive leading coefficient ($�{�}^2+��+�$) opens upward and has a global minimum. A quadratic polynomial with a negative leading coefficient ($�{�}^2+��+�$) opens downward and has a global maximum.

4. Closed Intervals:

   • Even polynomials may have global extrema when restricted to a closed interval (e.g., $[�,�]$) where the polynomial is continuous. In such cases, you can apply the Extreme Value Theorem to determine the existence of global extrema.

In summary, whether a polynomial function has a global minimum or maximum depends on the degree, the leading coefficient, and the interval over which the function is defined. Odd-degree polynomials and even-degree polynomials with certain characteristics do not have global extrema, while even-degree polynomials may have them, especially when constrained to a closed interval.