Question

Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? f(x)=1+3x 2 −2x 3

Answer 1

To find the local maximum and minimum values of the function f(x) = 1 + 3x^2 - 2x^3 using the First and Second Derivative Tests, follow these steps: find the first derivative, find the second derivative, apply the First Derivative Test, and apply the Second Derivative Test.

Step-by-step explanation:

To find the local maximum and minimum values of the function f(x) = 1 + 3x^2 - 2x^3 using the First and Second Derivative Tests, we need to follow these steps:

1. Find the first derivative of f(x) to find the critical points where f'(x) = 0 or f'(x) is undefined. These points could be possible local maximum or minimum points.
2. Find the second derivative of f(x) to determine the concavity of the function at the critical points found in step 1.
3. Apply the First Derivative Test to analyze the critical points. If the first derivative changes from positive to negative at a critical point, it is a local maximum. If the first derivative changes from negative to positive, it is a local minimum.
4. Apply the Second Derivative Test to analyze the critical points. If the second derivative is positive at a critical point, it is a local minimum. If the second derivative is negative, it is a local maximum.

The preferred method depends on the specific problem and personal preference. Both methods can be used to find the local maximum and minimum values of a function.