How to maximize $x^2-2bxy-7x+2y^2+133$?

Question

Suppose you have some function $g_b(x,y)=x^2-2bxy-7x+2y^2+133$ where $b\in\mathbb{R}$.

How would you find the maximum of $g_b(x,y)$ on the set $\{(x,y)\in\mathbb{R}^2\ |\ x^2+3y^2\leq 16\}$?

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My intuition suggests that I should factor the function such that for any $(x,y)$, it is easy to identify which ones are positive or negative (i.e. use the fact that $x^2\geq 0$).

However, I honestly don't know if that is the correct direction since I can't figure out how to factor this with $b$ unknown. I appreciate any guidance! Thank you!

Answer 1

Hint:

Let $g_b(x, y)=a$

$x^2-2bxy-7x+2y^2+133=a$

Solve equation for x:

$x=(1/2)[2by+7 + \sqrt \Delta]$ $\Delta=(2by+7)^2-4(2y^2+133-a)=(4b^2-8)y^2+2by-483+4a$

Now x is maximum if $\Delta$ is maximum. For this take derivative of $\Delta$ equate it to zero and so on.