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

Derive the Schwarz inequality using the identity above.

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TopNewest@Ishan Singh can you post a derivation? – Hummus A · 1 year, 7 months ago

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– Ishan Singh · 1 year, 7 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.Log in to reply