Let $f(x)=e^{3 x}-kx$, for $k>0$.

Using a calculator or computer, sketch the graph of $f$ for $k=1/9, 1/6,1/3,1/2,1,2,4$. Describe what happens as $k$ changes.

$f(x)$ has a local minimum. Find the location of the minimum.
$x =$

Find the $y$-coordinate of the minimum.
$y =$

Find the value of $k$ for which this $y$-coordinate is largest.
$k =$

How do you know that this value of $k$ maximizes the $y$-coordinate? Find $d^2y/dk^2$ to use the second-derivative test.
${d^2y\over dk^2} =$
(Note that the derivative you get is negative for all positive values of $k$, and confirm that you agree that this means that your value of $k$ maximizes the $y$-coordinate of the minimum.)