I need help understanding what this question is asking and I am not sure what to do. The question is given below
For each number y find the maximum value of $yx - 2x^4$. This maximum is a function $G(y)$. Verify that the derivative $G(y)$ and $2x^4$ are inverse functions
Please do not give away the answer but guide me in the right direction.
Many thanks and stay safe!!!
For the first part, it wants you to treat $y$ as a constant. For instance, if $y=2$, you would need to find the maximum of the function $2x-2x^4$ with respect to $x$. Finding the maximum for general $y$ gives you a function $G(y)$ such that $yG(y)-2(G(y))^4\geq yx-2x^4$ for any $x$.
As an example, if you wanted to find the maximum of $x-yx^2$ with respect to $x$, you would first find where the derivative is $0$, so $$0=\frac{d}{dx}x-yx^2=1-2yx$$ Solving for $x$, you get the function $\frac{1}{2y}$, which is what they mean by $G(y)$ in the problem (you still need to verify this is actually a maximum of course).
The second part is worded a little strangely, but I think they want you to verify that the inverse function of $G$ is equal to the derivative of $2x^4$, although I’m not sure why they would ask for this without more context, so I might be misreading it.