Question

Given the graph of f ′(x), the derivative of f(x), which of the following statements are true about the graph of f(x)?

Graph is an inverted parabola with x intercepts at x equals 0 and 4 and vertex at x equals 2.


The graph of f does not have a point of inflection.

The graph of f has a point of inflection at x = 2.

The graph of f has a critical point at x = 2.

I only

II only

III only

I and III only

Answer 1

$f'$ has a horizontal tangent line at $x=2$, and it's clear from the plot that $f'$ is increasing to the left and descreasing to the right. This means $f''=0$ when $x=2$, so there is an inflection point there and only statement II is true.

Statement III is not true because critical points only occur where $f'=0$ or (possibly) when $f'$ is undefined. Clearly $f'\neq0$ when $x=2$.