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) –4 –2 0 4 5

If f ?(x) is always increasing, which statement about f(x) must be true?
a) f(x) passes through the origin
b) f(x) is concave downwards for all x.
c) f(x) has a relative minimum at x = 0.
d) f(x) has a point of inflection at x = 0.

Answer 1

c) f(x) has a relative minimum at x = 0.

Explanation:

A differentiable function will have a relative minimum where the derivative is zero and the second derivative is positive.

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Here, the derivative f'(x) is increasing for all of the x-values shown in the table, so we can assume the second derivative is positive at those values.

The derivative is zero and the second derivative is positive at x=0, so ...

c) f(x) has a relative minimum at x = 0