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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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