Please help with a step by step solution to this problem **Question 3:** Determine where \( f(x) = 12x - x^3 \) has any local maximums or any local minimums.**Analysis:** To find the local maximums and minimums, we need to compute the first derivative of the function \( f(x) \) and then find the critical points by setting the derivative equal to zero.1. **First Derivative:** \[ f'(x) = \frac{d}{dx}(12x - x^3) = 12 - 3x^2 \]2. **Critical Points:** Set \( f'(x) = 0 \) to find critical points: \[ 12 - 3x^2 = 0 \] \[ 3x^2 = 12 \] \[ x^2 = 4 \] \[ x = \pm 2 \]3. **Second Derivative Test:** To determine whether these critical points are local maximums or minimums, use the second derivative: \[ f''(x) = \frac{d}{dx}(-3x^2) = -6x \] - Evaluate \( f''(x) \) at the critical points: - At \( x = 2 \): \[ f''(2) = -6(2) = -12 \quad (\text{Negative, so } x = 2 \text{ is a local maximum}) \] - At \( x = -2 \): \[ f''(-2) = -6(-2) = 12 \quad (\text{Positive, so } x = -2 \text{ is a local minimum}) \]**Conclusion:**The function \( f(x) = 12x - x^3 \) has a local maximum at \( x = 2 \) and a local minimum at \( x = -2 \).

Answered: 3. Determine where f(x) = 12x - x³ has…

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3. Determine where f(x) = 12x - x³ has any local maximums or any local minimums.

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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