a) Let $n \in \mathbb{N}^*$ and $c \in \mathbb{R}_+$. Find all continuous functions $f : [0,1] \to \mathbb{R}_+$ such that $$ \int_0^1{(f(t))^ndt} = \int_0^1{(f(t))^{n+1}dt} = \int_0^1{(f(t))^{n+2}dt} = c .$$
b) Let $n \in \mathbb{N}^*$ and $c \in \mathbb{R}_+$. Find all continuous functions $f : [0,1] \to \mathbb{R}_+$ such that $$ \int_0^1{(f(t))^ndt} = \int_0^1{(f(t))^{n+1}dt} = c .$$
I just can't get a grasp on this problem. I don't want a full response, just a hint that will help me.
a) Since $f$ is positive and continuous, $$ \int_{0}^{1}f(x)^n\,dx \int_{0}^{1}f(x)^{n+2}\,dx \geq \left(\int_{0}^{1}f(x)^{n+1}\,dx\right)^2 $$ by the Cauchy-Schwarz inequality, and equality is achieved iff $f(x)$ is constant.
In particular the only solution is $f(x)=1$ for $c=1$.
b) We cannot have $f(x)<1$ or $f(x)>1$ for any $x\in[0,1]$ since otherwise the equality between the $n$-th moment and the $(n+1)$-th moment would be violated. So, let $A$ be the subset of $[0,1]$ over which $f(x)\leq 1$ and let $B$ the subset of $[0,1]$ over which $f(x)\geq 1$: they both are given by a non-empty closed set and $\mu(A)+\mu(B)=1$. On the other hand, by the same argument as above the function $g:\mathbb{N}\to \mathbb{R}^+$ given by $$ g(m)=\log\int_{0}^{1}f(x)^m\,dx $$ is convex and $g(0)=0$ implies $c\leq 1$. Can we build discontinuous solutions given by $f(x)=a<1$ over $\left[0,\frac{1}{2}\right]$ and $f(x)=b>1$ over $\left(\frac{1}{2},1\right]$, for a suitable choice of $a$ and $b$?