Find the exact location of all the relative and absolute extrema of the function. f(x) = Vx(x + 5); x 2 0 The variable f has an absolute minimum v at (x, y) =
Find the exact location of all the relative and absolute extrema of the function. f(x) = Vx(x + 5); x 2 0 The variable f has an absolute minimum v at (x, y) =
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Finding Extrema of Functions
**Problem:**
Find the exact location of all the relative and absolute extrema of the function.
\[
f(x) = \sqrt{x(x + 5)}, \quad x \geq 0
\]
**Solution:**
The variable \( f \) has:
- (Dropdown selection) "an absolute minimum" ✔
- at \( (x, y) = \left( \boxed{} \right) \).
**Explanation:**
Given the function \( f(x) = \sqrt{x(x + 5)} \) for \( x \geq 0 \), you need to determine where the function reaches its minimum or maximum values.
**Step-by-Step:**
1. **Find the Domain:**
- The function is defined for \( x \geq 0 \) because it involves a square root and multiplying by \( x \) which should be non-negative.
2. **Identify Critical Points:**
- Differentiate the function \( f(x) \).
- Set the derivative equal to zero to find critical points.
3. **Evaluate End Points and Critical Points:**
- Compute the function values at critical points.
- Evaluate the endpoints if necessary.
4. **Compare Values for Extrema:**
- Determine the minimum and maximum values by comparing values from step 3.
\[
f(x) = \sqrt{x(x + 5)}
\]
In this step, let's compute the derivative \( f'(x) \) and set it to zero to find critical points, if any. With the critical points, you can find the precise coordinates where the function is minimized or maximized.
Fill in the coordinates \( (x, y) \) where \( f \) has its absolute minimum.
This process helps in understanding how to find relative and absolute extrema for functions defined within specific domains.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7856584-59fd-4634-ab45-c4c5f6444c59%2F34250417-9eb5-4d11-908c-18a31cba7614%2Fe2rfyqq_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding Extrema of Functions
**Problem:**
Find the exact location of all the relative and absolute extrema of the function.
\[
f(x) = \sqrt{x(x + 5)}, \quad x \geq 0
\]
**Solution:**
The variable \( f \) has:
- (Dropdown selection) "an absolute minimum" ✔
- at \( (x, y) = \left( \boxed{} \right) \).
**Explanation:**
Given the function \( f(x) = \sqrt{x(x + 5)} \) for \( x \geq 0 \), you need to determine where the function reaches its minimum or maximum values.
**Step-by-Step:**
1. **Find the Domain:**
- The function is defined for \( x \geq 0 \) because it involves a square root and multiplying by \( x \) which should be non-negative.
2. **Identify Critical Points:**
- Differentiate the function \( f(x) \).
- Set the derivative equal to zero to find critical points.
3. **Evaluate End Points and Critical Points:**
- Compute the function values at critical points.
- Evaluate the endpoints if necessary.
4. **Compare Values for Extrema:**
- Determine the minimum and maximum values by comparing values from step 3.
\[
f(x) = \sqrt{x(x + 5)}
\]
In this step, let's compute the derivative \( f'(x) \) and set it to zero to find critical points, if any. With the critical points, you can find the precise coordinates where the function is minimized or maximized.
Fill in the coordinates \( (x, y) \) where \( f \) has its absolute minimum.
This process helps in understanding how to find relative and absolute extrema for functions defined within specific domains.
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