Let $f$ be a twice-differentiable function such that $f ( 2 ) = 5$ and $f ( 5 ) = 2$. Let $g$ be the function given by $g ( x ) = f ( f ( x ) )$. (a) Explain why there must be a value $c$ for $2 < c < 5$ such that $f ^ { \prime } ( c ) = - 1$. (b) Show that $g ^ { \prime } ( 2 ) = g ^ { \prime } ( 5 )$. Use this result to explain why there must be a value $k$ for $2 < k < 5$ such that $g ^ { \prime \prime } ( k ) = 0$. (c) Show that if $f ^ { \prime \prime } ( x ) = 0$ for all $x$, then the graph of $g$ does not have a point of inflection. (d) Let $h ( x ) = f ( x ) - x$. Explain why there must be a value $r$ for $2 < r < 5$ such that $h ( r ) = 0$.
Let $f$ be a twice-differentiable function such that $f ( 2 ) = 5$ and $f ( 5 ) = 2$. Let $g$ be the function given by $g ( x ) = f ( f ( x ) )$.
(a) Explain why there must be a value $c$ for $2 < c < 5$ such that $f ^ { \prime } ( c ) = - 1$.
(b) Show that $g ^ { \prime } ( 2 ) = g ^ { \prime } ( 5 )$. Use this result to explain why there must be a value $k$ for $2 < k < 5$ such that $g ^ { \prime \prime } ( k ) = 0$.
(c) Show that if $f ^ { \prime \prime } ( x ) = 0$ for all $x$, then the graph of $g$ does not have a point of inflection.
(d) Let $h ( x ) = f ( x ) - x$. Explain why there must be a value $r$ for $2 < r < 5$ such that $h ( r ) = 0$.