Apentix/AP Calculus AB/Unit 5: Analytical Applications of Differentiation/Question

AP Calculus ABhardmcq1 pt

Let $f'(x)=x^2(x-3)$ . Which conclusion is correct?

A.

f

has a local minimum at

x=3

, and the second derivative test is inconclusive at

x=0

.

B.

f

has a local maximum at

x=3

.

C.

f

has a local minimum at

x=0

.

D.

f

has a local maximum at

x=0

and a local minimum at

x=3

.

Explanation

Core Concept

$f''(x)=3x(x-2)$ , so $f''(3)=9>0$ and $x=3$ is a local minimum. At $x=0$ , $f''(0)=0$ and $f'$ does not change sign, so the second derivative test is inconclusive.

Correct Answer

A $f$ has a local minimum at $x=3$ , and the second derivative test is inconclusive at $x=0$ .

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