Question

Various values for the derivative, f'(x), of a differentiable function f are shown below.

x=1, f'(x) = 8
x=2, f'(x) = 4
x=3, f'(x) = 0
x=4, f'(x) = -4
x=5, f'(x) = -8
x=6, f'(x) = -12

If f'(x) always decreases, then which of the following statements must be true?
A) f(x) has a relative minimum at x=3
B) f(x) is concaved upward for all x
C) The graph of f(x) is symmetric with the line x=3
D) f(x) changes concavity at x=3
E) f(x) has a relative maximum at x=3

Answer 1

It has to be A or E. You have relative maxima or minima (or an inflection point) when the derivative is zero

(A) can be ruled out because the second derivative is negative at x = 3.

What choice does that leave you with?