MHT CET · Maths · Application of Derivatives
The function, \(f(x)=x \sqrt{1-x}\), where \(x \in(0,1)\), has local maximum at \(x=\)
- A \(\frac{1}{3}\)
- B \(\frac{1}{4}\)
- C \(\frac{2}{3}\)
- D \(\frac{3}{4}\)
Answer & Solution
Correct Answer
(C) \(\frac{2}{3}\)
Step-by-step Solution
Detailed explanation
\(f(x)=x \sqrt{1-x}\)
\(\Rightarrow f^{\prime}(x)=1 \sqrt{1-x}+\frac{x}{2 \sqrt{1-x}} x(-1)=\) \(\frac{2-3 x}{2 \sqrt{1-x}}\)
sign scheme of \(f^{\prime}(x)\)
Hence, local maxima at \(x=\frac{2}{3}\)
\(\Rightarrow f^{\prime}(x)=1 \sqrt{1-x}+\frac{x}{2 \sqrt{1-x}} x(-1)=\) \(\frac{2-3 x}{2 \sqrt{1-x}}\)
sign scheme of \(f^{\prime}(x)\)
Hence, local maxima at \(x=\frac{2}{3}\)
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