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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Application of Derivatives

NDA 2017 · Mathematics · Application of Derivatives

The maximum value of In x x is

The maximum value of \frac{\operatorname{In} x}{x}$ is

Options

  1. Ae{e}
  2. B1e\frac{1}{\mathrm{e}}
  3. C2e\frac{2}{\mathrm{e}}
  4. D11

Correct answer

B. 1e\frac{1}{\mathrm{e}}

Step-by-step solution

f(x)=ℓnxx\mathrmf′(x)=(1x)x−ℓnxx2=1−ℓnxx2\begin{aligned} & \mathrm{f}(\mathrm{x})=\frac{\ell \mathrm{nx}}{\mathrm{x}} \\ & \mathrmf^{\prime}(\mathrm{x})=\frac{\left(\frac{1}{\mathrm{x}}\right) \mathrm{x}-\ell \mathrm{nx}}{\mathrm{x}^2}=\frac{1-\ell \mathrm{nx}}{\mathrm{x}^2}\end{aligned}\mathrmf′(x)=0⇒1−ℓnxx2=0⇒1−ln⁡x=0⇒ln⁡x=1⇒x=e\begin{aligned} & \mathrmf^{\prime}(\mathrm{x})=0 \Rightarrow \frac{1-\ell \mathrm{nx}}{\mathrm{x}^2}=0 \Rightarrow 1-\ln \mathrm{x}=0 \\ & \Rightarrow \ln x=1 \\ & \Rightarrow \mathrm{x}=\mathrm{e}\end{aligned}Hence,maximumvalueofHence, maximum value off(x)=\frac{1}{e}$
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