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…

Math

Calculus

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...

See similar textbooks

Similar questions

1) b) Solve for x
Decide if each of the following problems is a linearization problem, an optimization prob- lem, a related rates problem, a graphing problem, or a Newton's method problem. Identify. Label the problem as one of the five types or explain that it is not exactly one of those types (and how you know). Begin solution. For each problem, start the solution.
You must complete the two "designated for crediť" problems. In order to receive consideration for fu credit, all work must be shown. This includes computations, graphs, and explanations, when directec do so. The credited CTPS are worth a combined total of 25 points. You will be awarded points based accuracy and effort. Completely documenting your work increases my ability to rate your effort CTPS for Chapter 8 are due on Monday, April 25th. Included in this assignment are two extra credit problems. By completing the Extra Credit CTPS for Chapter 8 you will have an opportunity to earn up to 4 extra credit points per problem. All work, explanations, and correct answers must be provided in order for the problem to be considered fo extra credit. Students may turn in CTPS beyond the due date. No extra credit will be awarded if the assignment is submitted after the due date stated above. Credited Critical Thinking Problems 1. (Section 8.1) (10 points) In surveying, it is often necessary to…
Please show a step-by-step solution. Do not skip steps, and explain your steps. Write it on paper, preferably. Make sure the work is clear.
Can you help me on problem 2 A-C
A tortoise and hare are competing in a 400-meter race. The arrogant hare gives the tortoise a 220-meter head start. When the start gun is fired, the hare begins running at a constant speed of 3.5 meters per second and the tortoise begins crawling at constant speed of 1.5 meters per second. Take out a piece of paper and read the above problem context again. You will prepare your written work and solutions on your own paper and upload it and the end of this question. Complete the problem solving process by: i. reading and re-reading the problem to identify the quantities in the situation; ii. making a drawing to represent the relevant quantities in the situation; iii. and defining the variable t to represent the number of seconds since the start of the race. a. Define a function f to determine the distance of the tortoise from the finish line in terms of the number of seconds, t, since the start of the race. Preview Solve f(t) = 0 and describe what your solution represents. t = Preview…
Among the assumptions for the formulation of an LP problem, there us linearity. What does it mean in practice, and why is it important?
I keep getting these last two parts of this problem wrong, could someone show me how to solve it, please? (also, I know the third option for multiple choice isn't correct)