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

AP Calculus ABhardmcq1 pt

Let $f'(x)=(x-1)^2(x-4)$ . Which statement about $f$ is true?

A.

f

has a local maximum at

x=1

and a local minimum at

x=4

.

B.

f

is increasing on

(-\infty,4)

.

C.

f

has a local minimum at

x=4

but no local extremum at

x=1

.

D.

f

has a local maximum at

x=4

.

Explanation

Core Concept

$(x-1)^2$ is nonnegative, so $f'$ has the sign of $(x-4)$ : negative on $(-\infty,4)$ and positive on $(4,\infty)$ . Thus $f$ has a local minimum at $x=4$ , while $f'$ never changes sign at $x=1$ .

Correct Answer

C $f$ has a local minimum at $x=4$ but no local extremum at $x=1$ .

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