Optimization Question - Distance to the Origin

Question

I came across a question that asked what the furthest point from the origin subject to a constraint ie. $x^4+y^4+3xy=2$. It was an optimization question and we were told to use Lagrangian Multiplier. In the answer, they made the objective function $x^{2}+y^{2}$ but shouldn't it be $\sqrt{x^2+y^2}$? If not, why?

Answer 1

Because $\sqrt{z}$ is an increasing function of $z$, optimizing $\sqrt{x^2+y^2}$ is equivalent to optimizing $x^2+y^2$. The same approach is often used with $\log(z)$ as the increasing function instead of $\sqrt{z}$, especially for maximizing likelihood, which is usually expressed as a product. In that case, the motivation is that $\log$ transforms products to sums, and it is generally easier to differentiate sums.