The function, $f(x)=x \sqrt{1-x}$, where $x \in(0,1)$, has local maximum at $x=$

• A

  $\frac{1}{3}$
• B

  $\frac{1}{4}$
• Correct answer
  C

  $\frac{2}{3}$
• D

  $\frac{3}{4}$

Solution

$\begin{aligned} & f(x)=x \sqrt{1-x} \\ & \Rightarrow f^{\prime}(x)=1 \sqrt{1-x}+\frac{x}{2 \sqrt{1-x}} x(-1)=\frac{2-3 x}{2 \sqrt{1-x}} \end{aligned}$ sign scheme of $f^{\prime}(x)$
Hence, local maxima at $x=\frac{2}{3}$

Hence, the correct answer is option: 'C'.