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# Write the Point Where F(X) = X Log, X Attains Minimum Value. - Mathematics

#### Question

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

#### Solution

$\text{ Given:} \hspace{0.167em} f\left( x \right) = x \log_e x$

$\Rightarrow f'\left( x \right) = \log_e x + 1$

$\text{ For a local maxima or a local minima, we must have }$

$f'\left( x \right) = 0$

$\Rightarrow \log_e x + 1 = 0$

$\Rightarrow \log_e x = - 1$

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

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

$\text { Now,}$

$f''\left( x \right) = \frac{1}{x}$

$\text { At x } = \frac{1}{e}:$

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

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

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