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

AP Calculus ABhardmcq1 pt

Let $f(x)=x^4-4x^3$ . Which statement about the graph of $f$ is true?

A.

f

has a local maximum at

x=0

and a local minimum at

x=3

.

B.

f

has a point of inflection only at

x=2

.

C.

f

has critical points at

x=0

and

x=2

, and a point of inflection at

x=3

.

D.

f

has a local minimum at

x=3

and points of inflection at

x=0

and

x=2

.

Explanation

Core Concept

$f'(x)=4x^2(x-3)$ changes from negative to positive only at $x=3$ , a local minimum. $f''(x)=12x(x-2)$ changes sign at $x=0$ and $x=2$ , so both are points of inflection.

Correct Answer

D $f$ has a local minimum at $x=3$ and points of inflection at $x=0$ and $x=2$ .

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