Question

suppose that the function graphed below is f(x), the derivative of f(x). find the locations of all relative extrema of f(x), and tell whether each extremum is a relative maximum or minimum. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the function f(x) has a relative minimum at x = and has a relative maximum at x = (round to the nearest integer as needed. use a comma to separate answers as needed.) b. the function f(x) has a relative minimum at x = and has no relative maximum. (round to the nearest integer as needed. use a comma to separate answers as needed.) c. the function f(x) has a relative maximum at x = and has no relative minimum. (round to the nearest integer as needed. use a comma to separate answers as needed.) d. the function f(x) has no relative extrema. e. there is not enough information given.

Explanation:

Step1: Recall Extrema from Derivative

To find relative extrema of \( f(x) \), we use the First Derivative Test: relative minima occur where \( f'(x) \) changes from negative to positive, and relative maxima occur where \( f'(x) \) changes from positive to negative.

Step2: Analyze \( f'(x) \) Graph

• Look at the graph of \( f'(x) \). Find where \( f'(x) \) crosses the x - axis (critical points) and check the sign change:
• At \( x=-8 \): Check the sign of \( f'(x) \) around \( x = - 8 \). To the left of \( x=-8 \), \( f'(x) \) is negative (since the graph is below the x - axis), and to the right of \( x=-8 \), \( f'(x) \) becomes positive (graph above the x - axis). So, \( f'(x) \) changes from negative to positive at \( x=-8 \), which means \( f(x) \) has a relative minimum at \( x=-8 \).
• At \( x=-4 \): To the left of \( x = - 4 \), \( f'(x) \) is positive (graph above x - axis), and to the right of \( x=-4 \), \( f'(x) \) becomes negative (graph below x - axis). So, \( f'(x) \) changes from positive to negative at \( x=-4 \), which means \( f(x) \) has a relative maximum at \( x=-4 \).

Answer:

A. The function \( f(x) \) has a relative minimum at \( x=-8 \) and has a relative maximum at \( x=-4 \).