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Write the point where

Question:

Write the point where $f(x)=x \log , x$ attains minimum value.

Solution:

Given : $f(x)=x \log _{e} x$

$\Rightarrow f^{\prime}(x)=\log _{e} x+1$

For a local maxima or a local minima, we must have

$f^{\prime}(x)=0$

$\Rightarrow \log _{e} x+1=0$

$\Rightarrow \log _{e} x=-1$

$\Rightarrow x=\frac{1}{e}$

$\Rightarrow f\left(\frac{1}{e}\right)=\frac{1}{e} \log _{e}\left(\frac{1}{e}\right)=-\frac{1}{e}$

Now,

$f^{\prime \prime}(x)=\frac{1}{x}$

$f^{\prime \prime}\left(\frac{1}{e}\right)=\frac{1}{\frac{1}{e}}=e>0$

So, $\left(\frac{1}{e},-\frac{1}{e}\right)$ is a point of local minimum.

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