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.

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

You can think of this problem like a hill. At the top of the hill the slope is zero, and on either side the slope has a different sign.

If we approaching the hill from the 'left' side, the slope is positive, but its value decreases to zero at the top of the hill.

As we continue to the 'right' side of the hill, the slope starts at zero and becomes increasingly negative.

Clearly A) is the right answer. B) cannot be correct because a hill is concave down; a valley is concave up. C) is likely not correct because the change in slope is always negative (monotonic). D) is likely not correct because although the derivative of f(x) is zero when x = 0, that only means that the slope of the function is zero when x = 0. We are not given any information about the value of the function at that point.