Question

Find the interval in which $f(x) = x + \frac{1}{x}$ is always increasing, $x \neq 0$.

Hint:

A function is increasing where its derivative is positive. Here, we solved for the derivative of $f(x)$ to find when it is greater than zero.

Answer 1

The derivative of $f(x) = x + \frac{1}{x}$ is $f'(x) = 1 - \frac{1}{x^2}$. For the function to be increasing, $f'(x)>0$, which means $1 - \frac{1}{x^2}>0$. This simplifies to $\frac{1}{x^2}<1$, or $x^2>1$. Therefore, the function is increasing when $|x|>1$, meaning $x>1$ or $x<-1$. The function increases on the intervals \( (-\infty, -1) \cup (1, \infty) \).