Let $f$ be the function given by $f ( x ) = \frac { \ln x } { x }$ for all $x > 0$. The derivative of $f$ is given by $f ^ { \prime } ( x ) = \frac { 1 - \ln x } { x ^ { 2 } }$. (a) Write an equation for the line tangent to the graph of $f$ at $x = e ^ { 2 }$. (b) Find the $x$-coordinate of the critical point of $f$. Determine whether this point is a relative minimum, a relative maximum, or neither for the function $f$. Justify your answer. (c) The graph of the function $f$ has exactly one point of inflection. Find the $x$-coordinate of this point. (d) Find $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$.
Let $f$ be the function given by $f ( x ) = \frac { \ln x } { x }$ for all $x > 0$. The derivative of $f$ is given by $f ^ { \prime } ( x ) = \frac { 1 - \ln x } { x ^ { 2 } }$.\\
(a) Write an equation for the line tangent to the graph of $f$ at $x = e ^ { 2 }$.\\
(b) Find the $x$-coordinate of the critical point of $f$. Determine whether this point is a relative minimum, a relative maximum, or neither for the function $f$. Justify your answer.\\
(c) The graph of the function $f$ has exactly one point of inflection. Find the $x$-coordinate of this point.\\
(d) Find $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$.