<s>[INST] 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:$$a^3+b^3+c^3+d^3$$
 [/INST] Put $a^2=:p\geq0$, $\ b^2=:q\geq0$, $\ c^2=:r\geq0$, $\ d^2=:s\geq0$, and note that
$$(p+q)^{3/2}-p^{3/2}={3\over2}\int_p^{p+q} x^{1/2}\ dx\geq {3\over2}\int_0^q x^{1/2}\ dx=q^{3\over2}\ .$$
It follows that $p^{3/2}+q^{3\over2}\leq (p+q)^{3/2}$, so that we get
$$\eqalign{a^3+b^3+c^3+d^3 &\leq p^{3/2}+q^{3/2}+r^{3/2}+s^{3/2}\leq (p+q)^{3/2}+(r+s)^{3/2}\cr &\leq (p+q+r+s)^{3/2}= 4^{3/2}=8\ .\cr}$$
Putting $a=2$, $b=c=d=0$ the maximal possible value $8$ is indeed realized.
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