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Write the Point Where F(X) = X Log, X Attains Minimum Value. - CBSE (Science) Class 12 - Mathematics

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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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Solution Write the Point Where F(X) = X Log, X Attains Minimum Value. Concept: Graph of Maxima and Minima.
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