Let $f$ be a twice-differentiable function defined on the interval $- 1.2 < x < 3.2$ with $f ( 1 ) = 2$. The graph of $f ^ { \prime }$, the derivative of $f$, is shown above. The graph of $f ^ { \prime }$ crosses the $x$-axis at $x = - 1$ and $x = 3$ and has a horizontal tangent at $x = 2$. Let $g$ be the function given by $g ( x ) = e ^ { f ( x ) }$. (a) Write an equation for the line tangent to the graph of $g$ at $x = 1$. (b) For $- 1.2 < x < 3.2$, find all values of $x$ at which $g$ has a local maximum. Justify your answer. (c) The second derivative of $g$ is $g ^ { \prime \prime } ( x ) = e ^ { f ( x ) } \left[ \left( f ^ { \prime } ( x ) \right) ^ { 2 } + f ^ { \prime \prime } ( x ) \right]$. Is $g ^ { \prime \prime } ( - 1 )$ positive, negative, or zero? Justify your answer. (d) Find the average rate of change of $g ^ { \prime }$, the derivative of $g$, over the interval $[ 1,3 ]$.
Let $f$ be a twice-differentiable function defined on the interval $- 1.2 < x < 3.2$ with $f ( 1 ) = 2$. The graph of $f ^ { \prime }$, the derivative of $f$, is shown above. The graph of $f ^ { \prime }$ crosses the $x$-axis at $x = - 1$ and $x = 3$ and has a horizontal tangent at $x = 2$. Let $g$ be the function given by $g ( x ) = e ^ { f ( x ) }$.
(a) Write an equation for the line tangent to the graph of $g$ at $x = 1$.
(b) For $- 1.2 < x < 3.2$, find all values of $x$ at which $g$ has a local maximum. Justify your answer.
(c) The second derivative of $g$ is $g ^ { \prime \prime } ( x ) = e ^ { f ( x ) } \left[ \left( f ^ { \prime } ( x ) \right) ^ { 2 } + f ^ { \prime \prime } ( x ) \right]$. Is $g ^ { \prime \prime } ( - 1 )$ positive, negative, or zero? Justify your answer.
(d) Find the average rate of change of $g ^ { \prime }$, the derivative of $g$, over the interval $[ 1,3 ]$.