The graph of is smooth on and has horizontal tangents at and

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

The graph of

u=g(x)

is smooth on

0\le x\le6

and has horizontal tangents at

x=1

and

x=5

. The graph of

y=f(u)

is smooth on

0\le u\le4

and has a single critical point (horizontal tangent) at

u=2

. Both functions are otherwise strictly increasing wherever their derivatives are nonzero. Let

y=f(g(x))

0\le x\le6

. (a) Identify all

x

–values in

0\le x\le6

at which

\dfrac{du}{dx}=0

. Justify using the given information. (b) Determine every

x

–value in

0\le x\le6

at which

\dfrac{dy}{dx}=0

. Explain your reasoning carefully. (c) For each interval on which

\dfrac{dy}{dx}

is positive, state whether

y

is increasing or decreasing and justify your answer. (d) Suppose

g''(1)\ne0

and

f''(2)\ne0

. Discuss whether

x=1

x=5

could correspond to a local extremum of

y

, citing appropriate derivative tests.

Related theory

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