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

AP Calculus ABmediummcq1 pt

Let $f(x)=x^3-6x^2+9x+1$ . Which statement describes the critical points of $f$ ?

A.

f

has a local maximum at

x=1

and a local minimum at

x=3

.

B.

f

has a local minimum at

x=1

and a local maximum at

x=3

.

C.

f

has local maxima at both

x=1

and

x=3

.

D.

f

has no local extrema.

Explanation

Core Concept

$f'(x)=3x^2-12x+9=3(x-1)(x-3)$ . $f'$ is positive on $(-\infty,1)$ , negative on $(1,3)$ , and positive on $(3,\infty)$ , so $x=1$ is a local maximum and $x=3$ is a local minimum.

Correct Answer

A $f$ has a local maximum at $x=1$ and a local minimum at $x=3$ .

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