Question

F is a function that is differentiable for all reals. The value of f '(x) is given for several values of x in the table below.

x -8 -3 0 3 8
f '(x) 5 4 0 -2 -4
If f '(x) is always decreasing, which statement about f(x) must be true?


a.
f(x) has a relative maximum at x = 0.

b.
f(x) is concave upwards for all x.

c.
f(x) has a point of inflection at x = 0.

d.
f(x) passes through the origin.

Answer 1

9514 1404 393

Answer:

a. f(x) has a relative maximum at x = 0.

Explanation:

Since f' is decreasing, the second derivative is negative. That means the graph is concave downward, so any critical point is a relative maximum.

f(x) has a relative maximum at x=0 (where f'(0) = 0)

__

There is no indication that the second derivative is ever zero, so there is no point of inflection.

No values of f(x) are given, so we cannot say whether the function passes through the origin.