Suppose that the graph below is the graph of f'(x), the derivative of f(x). Find the locations of all relative extrema, and tell whether each extremum is a relative maximum or minimum. A 10- 6- 4 Select the correct choice below and fill in the answer box(es) within your choice. (Simplify your answer. Use a comma to separate answers as needed.) O A. The function f(x) has a relative minimum at x = and has no relative maximum OB. The function f(x) has a relative maximum at x = and has no relative minimum. OC. The function f(x) has a relative minimum at x = and a relative maximum at x = 7 OD. There is not enough information given. O E. The function f(x) has no relative extrema. 2- 10-8 24 68 10 -2 – 6 -10

EXPERT ANSWER

In this question given in above figure we use the first derivative test to find relative maximum and minimum.

If the derivative f’ (x) changes from positive (increasing function) to negative (decreasing function), the function has a local (relative) maximum at the critical point. … When this technique is used to determine local maximum or minimum function values, it is called the First Derivative Test for Local Extrema.

Here at x=4 the derivative changes from -ve to + ve so at this point f(x) has relative minima.

at x=6 the derivative changes from +ve to -ve so at this point f(x) has relative Maxima.

Option (C) is correct.

The function f(x) has relative minima at x=4 and relative maxima at x=6.