Find minimum in a constrained two-variable inequation I would appreciate if somebody could help me with the following problem: 
Q: find minimum 
$$9a^2+9b^2+c^2$$
where $a^2+b^2\leq 9, c=\sqrt{9-a^2}\sqrt{9-b^2}-2ab$

Another approach - the symmetry here suggests Purkiss Principle (conditions to be verified), so the extremum is attained when $a = b$.  
So $c = (9-a^2) - 2a^2 = 9-3a^2$
and $9(a^2+b^2) + c^2 = 18a^2 + (9-3a^2)^2 = 9a^4 - 36a^2 + 81 = 9 (a^2 - 2)^2 + 45$  
which is minimised when $a^2 = 2$ or $a = \pm \sqrt2$.
