Question

How to Find the Critical Values of a Function

A: Derivative Test |
B: Second Derivative Test |
C: Critical Point Analysis |
D: Concavity Analysis

Answer 1

To find critical values of a function, you can use the derivative test, second derivative test, critical point analysis, and concavity analysis.

Step-by-step explanation:

How to Find the Critical Values of a Function:

1. Derivative Test: To find critical values using the derivative test, find the derivative of the given function and set it equal to zero. Solve the resulting equation to find the critical values. Use the first derivative test to determine if these values correspond to local maxima or minima.
2. Second Derivative Test: If the first derivative test is inconclusive, you can use the second derivative test. Find the second derivative of the function and evaluate it at the critical values. If the second derivative is positive, the critical value corresponds to a local minimum. If the second derivative is negative, it corresponds to a local maximum.
3. Critical Point Analysis: In addition to using the derivative and second derivative tests, you can also analyze critical points by determining if they occur at endpoints of the domain or points of discontinuity.
4. Concavity Analysis: To determine the concavity of a function, find the second derivative and analyze its sign changes. If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.