#### Question

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

#### Solution

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

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

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

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

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

\[ \Rightarrow x^2 = 1\]

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

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

\[ \Rightarrow x = - 1\]

\[\text { Now,} \]

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

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

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

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

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

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

Is there an error in this question or solution?

Solution Write the Maximum Value of F(X) = X + 1 X , X > 0 . Concept: Graph of Maxima and Minima.