AP Calculus AB / Differentiation Composite, Implicit, and Inverse Functions / Question No. 45

45. The composite function \( h \) is defined by: \[ h(x) = f[g(x)], \] where \( f \) and \( g \) are functions whose graphs are shown below. The graph of \( f \) has horizontal tangents at \( x = -2 \) and \( x = 1 \). The graph of \( g \) has horizontal tangents at \( x = -3, 0 \), and \( 2 \). The number of points on the graph of \( h \) where there are horizontal tangent lines is:

Answer is: option3

Solution:

Given the composite function: \[ h(x) = f(g(x)) \] We differentiate both sides: \[ h'(x) = f'[g(x)] \cdot g'(x) \]

We need to find all the values of \( x \) where \( h'(x) = 0 \).

\( g'(x) = 0 \) when \( x = -3, 0, \) and \( 2 \).
\( f'(x) = 0 \) when \( x = -2 \) and \( 1 \).

In order for \( f'[g(x)] = 0 \), we need \( g(x) = -2 \) or \( g(x) = 1 \).

From the given information, this occurs when:

\( x = -2, 0, \) and \( 3.5 \).

Thus, \( h'(x) = 0 \) when \( x = -3, 0, 2, -2, \) and \( 3.5 \).

There are 5 values.

Final Answer: 5

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