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  4. If x and y are both non-negative and if x+y=π, then the maximum value of 5 sin x sin y is equal to

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Q. If $x$ and $y$ are both non-negative and if $x+y=\pi$, then the maximum value of $5 \sin x \sin y$ is equal to

KEAMKEAM 2022

Solution:

$ f(x)=5 \sin x \cdot \sin y $
$ f(x)=5 \sin x \sin (\pi-x) $
$ =5 \sin x \cdot \sin x$
$ f(x)=5 \sin ^2 x $
$f^{\prime}(x)=0 \sin 2 x=0 $
$ f^{\prime}(x)=0 \sin 2 x=0 $
$x=\pi / 2 $
$\Rightarrow 5$