$f:\mathbb{R}^2 \to \mathbb{R}$
$f((a,b)^T)=5a^2+7b^2-14ab$
How can I determine the extrema (minima and maxima) on $D:=\{(a,b)^T \in \mathbb{R}^2: \|(a,b)^T\|_2=1\}$?
My play is to write $b=\pm\sqrt{1-x^2}$ and then look where $f_a$ and $f_b$ are $0$ and then use the second derivative test.
But am I allowed to this? Because I am afraid I can't just differentiate on $D$ itself. But I would arrive at the correct result when I compare it to wolframalpha.
Can this be done like this or is this wrong and how else can it be done?
(We haven't heard about Lagrange multipliers yet if that makes any difference)