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# F(X) = X √ 1 − X , X > 0 . - Mathematics

#### Question

f(x) =$x\sqrt{1 - x} , x > 0$.

#### Solution

$\text { Given }: f\left( x \right) = x\sqrt{1 - x}$

$\Rightarrow f'\left( x \right) = \sqrt{1 - x} - \frac{x}{2\sqrt{1 - x}} = \frac{2 - 3x}{2\sqrt{1 - x}}$

$\text { For the local maxima or minima, we must have }$

$f'\left( x \right) = 0$

$\Rightarrow \frac{2 - 3x}{2\sqrt{1 - x}} = 0$

$\Rightarrow x = \frac{2}{3}$

Since,  f '(x) changes from positive to negative when x increases through $\frac{2}{3}$,  x = $\frac{2}{3}$ is a point of maxima.

The local maximum value of  f (x) at x = $\frac{2}{3}$ is given by $\frac{2}{3}\sqrt{1 - \frac{2}{3}} = \frac{2}{3\sqrt{3}} = \frac{2\sqrt{3}}{9}$

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