Find the minimum of $M$

Here is my solution.

Let us introduce $u,v$ such that $\cos = a/\sqrt{a^2+b^2},\ \cos v = c/\sqrt{c^2+d^2}$ . Clearly, the constraints $a,b,c,d>0$ impose the constraints $u,v\in [0,\pi/2]$ . Note that now one can write $M = \frac{a^3}{c}+\frac{b^3}{d}=\frac{\cos^3u}{\cos v}+\frac{\sin^3 u}{\sin v},$ which follows from the given constraint $c^2+d^2=(a^2+b^2)^3$ . Therefore, minimizing $M$ is equivalent to minimizing the above expression in $u,v$ which is an unconstrained optimization problem except for the trivial constraint $u,v\in [0,\pi/2]$ . The minima can be easily calculated to be the point where $u=v$ , where the value of $M$ becomes $\cos^2u+\sin^2u=1$ .