Max and min value of $f(x,y)$ $$f(x,y) = 2\cdot x\cdot y$$
$$x^2+y^2 \leq 4$$
I have no idea about this question.There is a region.How can I solve?
At first make two cases: 
$$x^2+y^2 &lt;4$$ 
this one is open hence you need the gradient to be zero there.

The second case is the boundary so 
$$x^2+y^2=4$$ 
Here take langrange multipliers
Hint :Use A.M - G.M inequality for $|x|,|y|$.

$\Rightarrow x^2+y^2\ge2|xy|$

$\Rightarrow 4\ge x^2+y^2\ge 2|xy|$

$\Rightarrow-4\le2xy\le 4$

Equality holds when $|x|=|y|=\sqrt{2}$