Show that $f$ gets a global minimum and a global maximum on $D$

Question

Let $D=\{(x,y)\in \mathbb{R}^2\mid x^2+y^2\leq 1\}$ and $f: D \rightarrow \mathbb{R}$ with $f(x,y)=3x^2-2xy+3y^2$.

(a) Show that $f$ gets a global minimum and a global maximum on $D$.

(b) Determine all points on $D$ where $f$ gets its global minimum and global maximum.

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For question (b) first I found the critical points on the boundary of the circle $x^2+y^2=1$ using Lagrange multipliers. Then I determined the critical points of $f(x,y)$ for the interior of the circle. Then combining all results I foundthe global minimum and the global maximum.

But how can we show (a) ? Could you give me a hint?

Answer 1

Continuous function attains maximum and minimum on a compact set.

You just have to verify that your problem satisfies these properties.