Compute the minimum value of $a^n + b^n + c^n$ subject to $a^2 + b^2 + c^2 = 1 $ 
Assume that $a,b,c$  are non-negative real numbers and $n$ is a natural number $n \ge 3$. What is $f(n)=$ the minimum value of $a^n + b^n + c^n$ ? 

I find ; 
$$f(3) = \frac{1}{\sqrt{3}}\qquad f(4) = \frac{1}{3}$$
then I guess 
$$f(n) = \left(\frac{1}{\sqrt{3}} \right)^n \times 3 $$ 
Is that true?

The power mean inequality gives $$\sqrt[n]{\frac{a^n+b^n+c^n}{3}} \geq \sqrt[2]{\frac{a^2+b^2+c^2}{3}}=\sqrt{\frac13}$$
with equality if $a=b=c$. Hence the minimal value is $3\left(\frac13\right)^{n/2}$, which is attained if $a=b=c=\left(\frac13\right)^{1/2}$. 
