How to solve this problem? This is one of the problems in my semester examination:
For $x,y\in\mathbb{R}^+,x^2+y^2=1$, find the maximum value of $M=\sqrt{x}+\sqrt{2y}$.
I know it can be solved using the Cauchy-Schwarz inequality, but I can't solve it despite much effort with this idea.
Any help will be much appreciated! Although the examination is now over, but a solution for this problem will be very useful to me.
Thank you in advance.