JEE Mains · Maths · STD 12 - 6. Application of derivatives
The local maximum value of the function \(f(x)=\left(\frac{2}{x}\right)^{x^{2}}, x>0\), is
- A \((2 \sqrt{\mathrm{e}})^{\frac{1}{\mathrm{e}}}\)
- B \(\left(\frac{4}{\sqrt{\mathrm{e}}}\right)^{\frac{\mathrm{e}}{4}}\)
- C \((\mathrm{e})^{\frac{2}{\mathrm{e}}}\)
- D \(1\)
Answer & Solution
Correct Answer
(C) \((\mathrm{e})^{\frac{2}{\mathrm{e}}}\)
Step-by-step Solution
Detailed explanation
\(f(x)=\left(\frac{2}{x}\right)^{x^{2}} ; x>0\) \(\ell n f(x)=x^{2}(\ell \ln 2-\ell n x)\) \(f^{\prime}(x)=f(x)\{-x+(\ell n 2-\ell n x) 2 x\}\) \(f^{\prime}(x)=\underbrace{f(x)}_{+} \cdot \underbrace{x}_{+} \underbrace{(2 \ell n 2-2 \ell n x-1)}_{g(x)}\)…
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