I have the following function: $f(x,y) = x(y^2-x^2)- \frac{(x^2 +y^2)^2}{2\rho}+\frac{3x^2(y^2-4x^2)}{\rho}$ where $\rho>0$ is a constant.
My goal is to find the maximum value of this function subject to the constraint $x^2 +y^2 \leq 1$.
What is the best way to solve this problem?
My thoughts:
- I think I can replace the second term $\frac{(x^2 +y^2)^2}{2\rho}$ with $\frac{1}{2\rho}$ since due to the constraint $x^2 +y^2 \leq 1$.
But, I am not sure how to proceed. Can someone show me the steps for solving this problem? What would be the easiest way to approach this?