(Solved): Let f(x)=e4xkx, for k>0. Using a calculator or computer, sketch the graph of f for k=91,61 ...

Let $f(x)=e^{4x}-kx$, for $k>0$. Using a calculator or computer, sketch the graph of $f$ for $k=\frac{1}{9},\frac{1}{6},\frac{1}{3},\frac{1}{2},1,2,4$. Describe what happens as $k$ changes. (1) $f(x)$ has a local minimum. Find the location of the minimum: $x=\frac{\ln \left(\frac{1}{1}\right)}{2}$ (2) Find the $y$-coordinate of the minimum $y=\frac{4\left(1-\ln \left(\frac{1}{1}\right)\right)}{2}?$ (3) Find the value of $k$ for which this $y$-coordinate is largest $k=2$ (4) How do you know that this value of $k$ maximizes the $y$-coordinate? Find $\frac{d^{2}y}{d^2}$ to use the second derivative test: $\frac{d^{2}y}{4k^2}=$ Important Note that the derivative you get is negative for all positive values of $k$, and make sure you understand that this means that your value of $k$ maximizes the $y$-coordinate of the minimum)