Derive it!

$\left [\displaystyle\int _{ a }^{ b }{ f(x)g(x)\quad dx } \right ]^{ 2 }=\displaystyle\int _{ a }^{ b }{ { f }^{ 2 }(x)dx\displaystyle\int _{ a }^{ b }{ { g }^{ 2 }(x)\quad dx-\frac { 1 }{ 2 } \displaystyle\int _{ a }^{ b }{ \displaystyle\int _{ a }^{ b }{ [f(x)g(y)-f(y)g(x)] } ^{ 2 }dxdy } } }$

Derive the Schwarz inequality using the identity above.

#Calculus

Easy Math Editor

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Comments

@Ishan Singh can you post a derivation?

Hamza A - 5 years, 3 months ago

There's nothing left to derive. Just see that square of the integral on the R.H.S. is $\geq 0$ and we are done.

Ishan Singh - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$