Question

If $f(x) = x + \frac{1}{x}, \, x \geq 1$, show that $f$ is an increasing function.

Hint:

To prove that a function is increasing, calculate the derivative. If the derivative is always positive (or zero), the function is increasing.

Answer 1

We are given the function: \[ f(x) = x + \frac{1}{x}, \quad x \geq 1. \] To prove that $f(x)$ is an increasing function, we will compute the derivative of $f(x)$: \[ f'(x) = \frac{d}{dx} \left( x + \frac{1}{x} \right) = 1 - \frac{1}{x^2}. \] For $f(x)$ to be increasing, we need: \[ f'(x) \geq 0. \] Thus, \[ 1 - \frac{1}{x^2} \geq 0 \quad \Rightarrow \quad x^2 \geq 1. \] Since $x \geq 1$, this condition is satisfied for all $x \geq 1$. Therefore, $f(x)$ is increasing on $[1, \infty)$.