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Find the critical point and determine if the function is increasing or decreasing on the given intervals. y = x2 – 8x², x > 0 (Use decimal notation. Give your answer to three decimal places.) critical point c =

Find the critical point and determine if the function is increasing or decreasing on the given intervals. y = x2 – 8x², x > 0 (Use decimal notation. Give your answer to three decimal places.) critical point c =

Calculus: Early Transcendentals
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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**Problem Statement:** Find the critical point and determine if the function is increasing or decreasing on the given intervals. \[ y = x^{9/2} - 8x^2, \quad x > 0 \] *(Use decimal notation. Give your answer to three decimal places.)* **Answer Box:** Critical point \( c = \) [Your Answer Here] --- **Explanation:** The problem requires identifying the critical point of the function \( y = x^{9/2} - 8x^2 \) when \( x > 0 \). A critical point occurs where the first derivative of the function is either zero or undefined. **Steps to Solve:** 1. **Differentiate the Function:** - Find \( \frac{dy}{dx} \) by differentiating \( y = x^{9/2} - 8x^2 \). 2. **Set the Derivative to Zero:** - Solve \( \frac{dy}{dx} = 0 \) to find the critical points. 3. **Analyze Intervals:** - Determine the behavior of the function (increasing or decreasing) in the intervals defined by the critical points. 4. **Provide the Critical Point:** - Calculate and present the critical point to three decimal places in the answer box provided.
Transcribed Image Text:**Problem Statement:** Find the critical point and determine if the function is increasing or decreasing on the given intervals. \[ y = x^{9/2} - 8x^2, \quad x > 0 \] *(Use decimal notation. Give your answer to three decimal places.)* **Answer Box:** Critical point \( c = \) [Your Answer Here] --- **Explanation:** The problem requires identifying the critical point of the function \( y = x^{9/2} - 8x^2 \) when \( x > 0 \). A critical point occurs where the first derivative of the function is either zero or undefined. **Steps to Solve:** 1. **Differentiate the Function:** - Find \( \frac{dy}{dx} \) by differentiating \( y = x^{9/2} - 8x^2 \). 2. **Set the Derivative to Zero:** - Solve \( \frac{dy}{dx} = 0 \) to find the critical points. 3. **Analyze Intervals:** - Determine the behavior of the function (increasing or decreasing) in the intervals defined by the critical points. 4. **Provide the Critical Point:** - Calculate and present the critical point to three decimal places in the answer box provided.
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