How to prove that a function has a maximum and a minimum

Question

Consider the function $f(x, y) = \sqrt{xy}$ on the domain: $$D = \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 2, x \geq 0, y \geq 0\}$$

How would you explain that this function has a maximum and minimum point without having to do any calculations? I found it hard to explain this since the function doesn't have an interval and the values of x and y both go to ∞

Answer 1

Since $f$ is continuous and $D$ is compact (it is a closed and bounded set of $\mathbb{R}^2$, $f$ must have a maximum and a mnimum.

Answer 2

Hint: Use that $$\sqrt{\frac{x^2+y^2}{2}}\geq \frac{x+y}{2}\geq \sqrt{xy}$$ And also $$\sqrt{xy}\geq 0$$