JEE Mains · Maths · STD 12 - 6. Application of derivatives
If the function \(f(x)=\left(\frac{1}{x}\right)^{2 x} ; x>0\) attains the maximum value at \(\mathrm{x}=\frac{1}{\mathrm{e}}\) then :
- A \(\mathrm{e}^\pi<\pi^{\mathrm{e}}\)
- B \(\mathrm{e}^{2 \pi}<(2 \pi)^{\mathrm{e}}\)
- C \(\mathrm{e}^\pi>\pi^{\mathrm{e}}\)
- D \((2 \mathrm{e})^\pi>\pi^{(2 \mathrm{e})}\)
Answer & Solution
Correct Answer
(C) \(\mathrm{e}^\pi>\pi^{\mathrm{e}}\)
Step-by-step Solution
Detailed explanation
Let \(y=\left(\frac{1}{x}\right)^{2 x}\) \( \ell \text { ny }=2 x \ell n\left(\frac{1}{x}\right) \) \( \ell n y=-2 x \ell n x \) \( \frac{1}{y} \frac{d y}{d x}=-2(1+\ell n x)\) for \(\mathrm{x}>\frac{1}{\mathrm{e}} \mathrm{f}^{\mathrm{n}}\) is decreasing so, \( \mathrm{e}<\pi \)…
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