Maximizing $3 x^2+2 \sqrt{2} x y$ with $x^4+y^4=1$ I want to have maxizing value of $3 x^2+2 \sqrt{2} x y$  when $x^4+y^4=1$, $x>0,y>0$.
How can I solve it.

here is one way to do this: let us parametrize the constraint $$x^4 + y^4 = 1, x > 0, y > 0 \text{ by } x = \sqrt{\cos t}, y = \sqrt{\sin t}, 0 < t < \pi/2.$$  we need to maximise $$ f = 3x^2 + 2\sqrt 2xy=3\cos t + 2\sqrt 2\sqrt{\cos t \sin t} = 3\cos t + 2\sqrt{\sin 2t}, 0 < t < \pi/2$$
taking the derivative of $f$ and seeting it to zero, you have $$f' = -3\sin t + 2\frac{\cos 2t}{\sqrt{\sin 2t}}$$
only critical number of $f$ is $t = 0.4636$ and this is a global max $4.4721$ for $f(t),\,  0 \le t \le \pi/2.$
