Maximum Holds When $a=b=c=d$ , Right?

Note : This is just a simple version of this problem .

Solution :

$a^4 \leq a^4 + b^4 + c^4 + d^4 = 16 \Rightarrow a \leq 2 \Rightarrow a^5 \leq 2a^4$ . Similarly, it can be proved that $b^5 \leq 2b^4, c^5\leq 2c^4$ and $d^5 \leq 2d^4$ . Adding these up, we get $a^5 + b^5 + c^5 + d^5 \leq 2(a^4 + b^4 + c^4 + d^4) = 32$ , and equality holds if and only if one of $a,b,c,d$ is $2$ and rest all are zero.

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[This problem is taken from the book "Inequalities" by Zdravako Cvetovski. It is a nice book, and I recommend it for beginners]