Finding the maximum and minimum points of a function

Question

Consider the function $f(x, y) =\sqrt{xy}$ on the domain: $$D =\{(x, y) ∈ \mathbb R^2 : x^2 + y^2 ≤ 5, x ≥ 0, y ≥ 0\}$$ In terms of finding the minimum point, I calculated the derivative of the function and made the derivative equal to $0.$ The outcome was that $x=0$ and $y=0$ (global min).

However, what's the approach to calculating the maximum point?

Answer 1

Since $D$ is a compact set, the function achieves its maximum and minimum, and it is or in a critical point, or at the boundary of the region.

So besides critical points, one has to do a search independently on the lines/curves
$x=0,$ $y=0$, $x^2+y^2=5$ (this latest with the change of coordinates to polar coordinates, or with the use of Lagrange multipliers), and compare the values with
$f(0,0), f(0,\sqrt 5), f(\sqrt 5,0).$
The largest is the global maximum of $f$ on $D.$

Answer 2

Use that $$\sqrt{\frac{x^2+y^2}{2}}\geq \frac{x+y}{2}\geq \sqrt{xy}$$