If $\displaystyle f: \mathbb{R}\mapsto\mathbb{R}$ is continuous, and $\displaystyle \int^1_0\left(f(x)\right)^2 dx = 3$.
Find the maximum value of $\displaystyle \int^1_0 x f(x) dx$
Please explain to me...
You are so nice^^
If $\displaystyle f: \mathbb{R}\mapsto\mathbb{R}$ is continuous, and $\displaystyle \int^1_0\left(f(x)\right)^2 dx = 3$.
Find the maximum value of $\displaystyle \int^1_0 x f(x) dx$
Please explain to me...
You are so nice^^
by Holder's inequality: $\displaystyle \int_0^1 xf(x) \ dx \leq \int_0^1 x|f(x| \ dx \leq \sqrt{\int_0^1 x^2 dx \cdot \int_0^1 (f(x))^2 dx} = 1.$ also if $\displaystyle f(x) = 3x,$ then $\displaystyle \int_0^1 (f(x))^2 dx = 3$ and $\displaystyle \int_0^1 xf(x) dx = 1.$ thus $\displaystyle \max_f \int_0^1 xf(x) \ dx = 1.$