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Please solve RD Sharma class 12 Chapter Maxima and Minima exercise Multiple choice question, question 3 maths textbook solution.

Answers (1)

Answer:   e    

Hint: For local maxima or minima, we must have f'(x) =0  .

Given: f(x)=\frac{x}{\log _e x}


We have,

\begin{aligned} &f(x)=\frac{x}{\log _{e} x} \\ &f^{\prime}(x)=\frac{\log _{e} x-1}{\left(\log _{e} x\right)^{2}} \end{aligned}

For maxima and minima f'(x)=0

\Rightarrow \frac{\log _e x-1}{(\log _ex)^2}=0

\Rightarrow \log _e x-1=0

\Rightarrow \log _e x=10

\Rightarrow x=e


\begin{aligned} &f^{\prime \prime}(x)=\frac{-1}{x\left(\log _{e} x\right)^{2}}+\frac{2}{x\left(\log _{e} x\right)^{3}} \\ &f^{\prime \prime}(e)=\frac{-1}{e}+\frac{2}{e}=\frac{1}{e}>0 \end{aligned}

So, x =e  is a point of local minima.

Minimum Value of f(e)=\frac{e}{\log _e e}=e

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