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

The function $f(x) = x + \frac{1}{x}$ is defined for $x \geq 1$. To demonstrate that $f(x)$ is an increasing function, we calculate its derivative: $f'(x) = \frac{d}{dx} \left( x + \frac{1}{x} \right) = 1 - \frac{1}{x^2}$. An increasing function requires its derivative to be non-negative, i.e., $f'(x) \geq 0$. This implies $1 - \frac{1}{x^2} \geq 0$, which simplifies to $x^2 \geq 1$. Given the domain $x \geq 1$, this condition is met for all $x$ in this domain. Consequently, $f(x)$ is an increasing function on the interval $[1, \infty)$.