First and Second Derivative Test Question 6

The graph of a twice-differentiable function $f$ is shown in the figure above. Which of the following is true?

f(3)\lt f'(3) \lt f''(3)

f'(3)\lt f''(3) \lt f(3)

f''(3)\lt f(3) \lt f'(3)

f''(3)\lt f'(3) \lt f(3)

Approach

Though we can't obtain the exact values of the first or second derivative, we can obtain the sign.

Since the graph of the equation is concave down, $f''(3)\lt 0$ .

Since the tangent line to the point has a positive slope, $f'(3)\gt 0$ .

The graph gives us the point $(3,0)$ , which means $f(3)=0$ .