Answer
(a) Point B
(b) Point E
(c) Point A
Work Step by Step
(a)
$\frac{dy}{dx}=f'\gt0$. This means that $f$ is increasing at this point. We can eliminate points A, E (where graph $f$ is decreasing) and D (where graph $f$ is neither increasing nor decreasing).
$\frac{d^2y}{dx^2}=f''\gt0$. This means $f$ is concave upward. This leads us to pick the point B, since $f$ is concave upward at B.
(b)
$\frac{dy}{dx}=f'\lt0$. This means that $f$ is decreasing at this point. We can eliminate points B, C (where graph $f$ is increasing) and D (where graph $f$ is neither increasing nor decreasing).
$\frac{d^2y}{dx^2}=f''\lt0$. This means $f$ is concave downward. This leads us to pick the point E, since $f$ is concave downward at E.
(c)
$\frac{dy}{dx}=f'\lt0$. This means that $f$ is decreasing at this point. We can eliminate points B, C (where graph $f$ is increasing) and D (where graph $f$ is neither increasing nor decreasing).
$\frac{d^2y}{dx^2}=f''\gt0$. This means $f$ is concave upward. This leads us to pick the point A, since $f$ is concave upward at A.
