I know that if domain of function is compact set I should look for point where gradients is $0$ because of Weierstrass theorem. But I do not know to tackle this kind of problem. I am not looking for solution but for advices and intuition on how to think to solve this problem.
2026-04-11 18:35:29.1775932529
find $\sup$ and $\inf$ of $\frac{x^2y}{x^4+x^2+y^2}$ on $\{(x,y):0< x \leq y\}$
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Start by looking at special cases, such as when $x=1$. That should tell you what the infimum is.
To find the supremum, you could view $x$ as a constant and compute the partial derivative with respect to $y$. What can you say about the critical point? Can you express it in terms of $x$? Does it occur at some $y \geq x$?