# find the max value of integral

• Oct 13th 2009, 06:32 AM
GTK X Hunter
find the max value of integral
If $f: \mathbb{R}\mapsto\mathbb{R}$ is continuous, and $\int^1_0\left(f(x)\right)^2 dx = 3$.
Find the maximum value of $\int^1_0 x f(x) dx$

You are so nice^^ (Rofl)
• Oct 13th 2009, 07:36 AM
NonCommAlg
Quote:

Originally Posted by GTK X Hunter
If $f: \mathbb{R}\mapsto\mathbb{R}$ is continuous, and $\int^1_0\left(f(x)\right)^2 dx = 3$.
Find the maximum value of $\int^1_0 x f(x) dx$

You are so nice^^ (Rofl)

by Holder's inequality: $\int_0^1 xf(x) \ dx \leq \int_0^1 x|f(x| \ dx \leq \sqrt{\int_0^1 x^2 dx \cdot \int_0^1 (f(x))^2 dx} = 1.$ also if $f(x) = 3x,$ then $\int_0^1 (f(x))^2 dx = 3$ and $\int_0^1 xf(x) dx = 1.$ thus $\max_f \int_0^1 xf(x) \ dx = 1.$
• Oct 13th 2009, 10:08 AM
Opalg
Quote:

Originally Posted by GTK X Hunter
If $f: \mathbb{R}\mapsto\mathbb{R}$ is continuous, and $\int^1_0\left(f(x)\right)^2 dx = 3$.
Find the maximum value of $\int^1_0 x f(x) dx$

Please explain to me... explain to me.. (Crying)

Hint: Did you ever come across the Cauchy–Schwarz inequality?
• Oct 13th 2009, 05:54 PM
GTK X Hunter
Quote:

Originally Posted by NonCommAlg
by Holder's inequality: $\int_0^1 xf(x) \ dx \leq \int_0^1 x|f(x| \ dx \leq \sqrt{\int_0^1 x^2 dx \cdot \int_0^1 (f(x))^2 dx} = 1.$ also if $f(x) = 3x,$ then $\int_0^1 (f(x))^2 dx = 3$ and $\int_0^1 xf(x) dx = 1.$ thus $\max_f \int_0^1 xf(x) \ dx = 1.$

How can you conclude that $\sqrt{\int_0^1 x^2 dx \cdot \int_0^1 (f(x))^2 dx} = 1.$
• Oct 14th 2009, 05:19 PM
Krizalid
first assumption was $\int^1_0\left(f(x)\right)^2 dx = 3,$ and $\int_0^1x^2\,dx=\frac13.$