Suppose that the function $f$ has a continuous second derivative for all $x$, and that $f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 3$, and $f ^ { \prime \prime } ( 0 ) = 0$. Let $g$ be a function whose derivative is given by $g ^ { \prime } ( x ) = e ^ { - 2 x } \left( 3 f ( x ) + 2 f ^ { \prime } ( x ) \right)$ for all $x$. (a) Write an equation of the line tangent to the graph of $f$ at the point where $x = 0$. (b) Is there sufficient information to determine whether or not the graph of $f$ has a point of inflection when $x = 0$ ? Explain your answer. (c) Given that $g ( 0 ) = 4$, write an equation of the line tangent to the graph of $g$ at the point where $x = 0$. (d) Show that $g ^ { \prime \prime } ( x ) = e ^ { - 2 x } \left( - 6 f ( x ) - f ^ { \prime } ( x ) + 2 f ^ { \prime \prime } ( x ) \right)$. Does $g$ have a local maximum at $x = 0$ ? Justify your answer.
Suppose that the function $f$ has a continuous second derivative for all $x$, and that $f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 3$, and $f ^ { \prime \prime } ( 0 ) = 0$. Let $g$ be a function whose derivative is given by $g ^ { \prime } ( x ) = e ^ { - 2 x } \left( 3 f ( x ) + 2 f ^ { \prime } ( x ) \right)$ for all $x$.\\
(a) Write an equation of the line tangent to the graph of $f$ at the point where $x = 0$.\\
(b) Is there sufficient information to determine whether or not the graph of $f$ has a point of inflection when $x = 0$ ? Explain your answer.\\
(c) Given that $g ( 0 ) = 4$, write an equation of the line tangent to the graph of $g$ at the point where $x = 0$.\\
(d) Show that $g ^ { \prime \prime } ( x ) = e ^ { - 2 x } \left( - 6 f ( x ) - f ^ { \prime } ( x ) + 2 f ^ { \prime \prime } ( x ) \right)$. Does $g$ have a local maximum at $x = 0$ ? Justify your answer.