Question

Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]

Answer 1

\[\text { Given }: \hspace{0.167em} f\left( x \right) = x + \frac{1}{x}\]

\[ \Rightarrow f'\left( x \right) = 1 - \frac{1}{x^2}\]

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

\[f'\left( x \right) = 0\]

\[ \Rightarrow 1 - \frac{1}{x^2} = 0\]

\[ \Rightarrow x^2 = 1\]

\[ \Rightarrow x = 1, - 1\]

\[\text { But }x > 0\]

\[ \Rightarrow x = 1\]

\[\text { Now,} \]

\[f''\left( x \right) = \frac{1}{x^3}\]

\[\text { At x} = 1: \]

\[f''\left( 1 \right) = \frac{2}{\left( 1 \right)^3} = 2 > 0\]

\[\text { So, x = 1 is a point of local minimum } . \]

\[\text { Thus, the local minimum value is given by }\]

\[f\left( 1 \right) = 1 + \frac{1}{1} = 1 + 1 = 2\]