The figure above shows the graph of $f ^ { \prime }$, the derivative of the function $f$, on the closed interval $- 1 \leq x \leq 5$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = 1$ and $x = 3$. The function $f$ is twice differentiable with $f ( 2 ) = 6$. (a) Find the $x$-coordinate of each of the points of inflection of the graph of $f$. Give a reason for your answer. (b) At what value of $x$ does $f$ attain its absolute minimum value on the closed interval $- 1 \leq x \leq 5$ ? At what value of $x$ does $f$ attain its absolute maximum value on the closed interval $- 1 \leq x \leq 5$ ? Show the analysis that leads to your answers. (c) Let $g$ be the function defined by $g ( x ) = x f ( x )$. Find an equation for the line tangent to the graph of $g$ at $x = 2$.
The figure above shows the graph of $f ^ { \prime }$, the derivative of the function $f$, on the closed interval $- 1 \leq x \leq 5$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = 1$ and $x = 3$. The function $f$ is twice differentiable with $f ( 2 ) = 6$.\\
(a) Find the $x$-coordinate of each of the points of inflection of the graph of $f$. Give a reason for your answer.\\
(b) At what value of $x$ does $f$ attain its absolute minimum value on the closed interval $- 1 \leq x \leq 5$ ? At what value of $x$ does $f$ attain its absolute maximum value on the closed interval $- 1 \leq x \leq 5$ ? Show the analysis that leads to your answers.\\
(c) Let $g$ be the function defined by $g ( x ) = x f ( x )$. Find an equation for the line tangent to the graph of $g$ at $x = 2$.