College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 84E
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Question
![### Finding the Relative Extreme Values of a Function
In this section, we will determine the relative extreme values of the polynomial function \( p(x) \). The given function is:
\[ p(x) = \frac{x^5}{5} - 3x^3 \]
**Steps to Find Relative Extremes:**
1. **Find the First Derivative:**
To find critical points, compute the first derivative of \( p(x) \):
\[ p'(x) = \frac{d}{dx} \left( \frac{x^5}{5} - 3x^3 \right) \]
2. **Set the First Derivative to Zero:**
Solve the equation \( p'(x) = 0 \) to find the critical points.
3. **Second Derivative Test:**
Compute the second derivative \( p''(x) \) and use it to determine whether each critical point is a local minimum, local maximum, or a saddle point.
By following these steps, one can locate the relative maximum or minimum points of this polynomial function.
**Breaking Down the Function:**
- The term \(\frac{x^5}{5}\) causes the function to grow very rapidly for large values of \(x\).
- The term \(-3x^3\) influences the fall and rise of the graph at smaller values of \(x\).
Analyzing relative extreme values allows us to understand the function's behavior and make insightful conclusions about its local maxima and minima.
For detailed graphs or practical examples, refer to the section on polynomial functions in the site’s Calculus area.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F111c8812-b57c-460c-bb56-120cfca2735f%2F5c033bfb-b5b1-47d1-a715-195877e5b766%2Fjbh2xik_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Relative Extreme Values of a Function
In this section, we will determine the relative extreme values of the polynomial function \( p(x) \). The given function is:
\[ p(x) = \frac{x^5}{5} - 3x^3 \]
**Steps to Find Relative Extremes:**
1. **Find the First Derivative:**
To find critical points, compute the first derivative of \( p(x) \):
\[ p'(x) = \frac{d}{dx} \left( \frac{x^5}{5} - 3x^3 \right) \]
2. **Set the First Derivative to Zero:**
Solve the equation \( p'(x) = 0 \) to find the critical points.
3. **Second Derivative Test:**
Compute the second derivative \( p''(x) \) and use it to determine whether each critical point is a local minimum, local maximum, or a saddle point.
By following these steps, one can locate the relative maximum or minimum points of this polynomial function.
**Breaking Down the Function:**
- The term \(\frac{x^5}{5}\) causes the function to grow very rapidly for large values of \(x\).
- The term \(-3x^3\) influences the fall and rise of the graph at smaller values of \(x\).
Analyzing relative extreme values allows us to understand the function's behavior and make insightful conclusions about its local maxima and minima.
For detailed graphs or practical examples, refer to the section on polynomial functions in the site’s Calculus area.
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