Question

The twice-differentiable functions f and g are defined for all real numbers x. Values of f, f ′, g, and g′ for various values of x are given in the table above.

Find the x-coordinate of each relative minimum of f on the interval [−2, 3 .] Justify your answers.

Answer 1

The x-coordinates of the relative minimums of f on the interval [-2, 3] are x = 0 and x = 2.

Step-by-step explanation:

To find the x-coordinate of each relative minimum of f on the interval [-2, 3], we need to identify the critical points of f and determine whether they are relative minimums or maximums. The critical points occur where f'(x) is equal to zero or undefined. Looking at the given table, we can see that f'(x) changes sign from positive to negative at x = 0 and from negative to positive at x = 2. Therefore, x = 0 and x = 2 are the x-coordinates of the relative minimums of f on the interval [-2, 3].