Provide an appropriate response.
-The critical values of are x = -2 and x = 2. Use the first derivative test to determine which of the critical values correspond to a local minimum.
A) x = 2 and x = -2
B) neither x = 2 nor x = -2 correspond to a local minimum
C) x = -2
D) x = 2

Question

Provide an appropriate response.
-The critical values of  are x = -2 and x = 2. Use the first derivative test to determine which of the critical values correspond to a local minimum.
A) x = 2 and x = -2 
B) neither x = 2 nor x = -2 correspond to a local minimum 
C) x = -2 
D) x = 2

Quizplus · Accepted Answer

The first derivative test involves evaluating the sign of the derivative to determine whether the critical point is a local maximum or a local minimum. If the sign changes from positive to negative at a critical point, then it is a local maximum. If the sign changes from negative to positive, then it is a local minimum. To use this test, we need to find the sign of the derivative on either side of each critical point. Since the critical values are x = -2 and x = 2, we need to evaluate the sign of the derivative for x < -2, -2 < x < 2, and x > 2. Based on this analysis, we find that the sign changes from positive to negative at x = 2, indicating that it is a local minimum. Therefore, the correct answer is D) x = 2. According to the first derivative test, if the derivative changes from negative to positive at a critical point, then it corresponds to a local minimum. Since the critical value of x = 2 is to the right of the other critical value x = -2 and the function changes from negative to positive at x = 2, it corresponds to a local minimum. Therefore, the correct answer is D.

Provide an Appropriate Response