Finding the maximum value If $a^2+b^2+c^2+d^2=4$ are real numbers, then how to find the maximum value for:$$a^3+b^3+c^3+d^3$$

About six years later... I just noticed the following: Observe that the given expression will attain its maximum when $a,b,c,d\geqslant 0$. Thus $$a^2+b^2+c^2+d^2=4\implies a^2\leqslant4\iff a\leqslant 2\iff a^2(a-2)\leqslant 0\iff \fbox{$\displaystyle a^3\leqslant 2a^2$}$$ $$\therefore \, a^3+b^3+c^3+d^3\leqslant 2a^2+2b^2+2c^2+2d^2=8$$
