Prove the inequality $(\frac{x+y}{2})^n≤\frac{x^n+y^n}{2}$ for every $x,y≥0$ and $n≥1$, by minimizing the function $f(x,y)=x^n+y^n$ subject to the constraint $x+y=a$
My attempt: want to minimizing $f(x,y)=x^n+y^n$ subject to the constraint $g(x,y,z)=x+y-a$. So $∇f=t∇g$ $(n\frac{x^n}{x},n\frac{y^n}{y}=t(1,1)$, then solving for t, I found $x=y$. Subbing this into g, I found $x=y=\frac{a}{2}$. Im unsure what I've done is correct. How should i approach this question?
Your approach is correct, $f(x,y) \geq f(\frac{a}{2}, \frac{a}{2})$ for $x + y = a$. So, $x^n + y^n \geq a^n / 2^{n-1}$, giving $${x^n + y^n \over 2} \geq \left(\frac a2\right)^n = \left(x+y\over2\right)^n$$