Which of these functions are strictly increasing on $(0, \infty)$?

I. $\log x$

II. $x + \frac{1}{x}$

III. $e^{-x}$

IV. $x^2$

Explanation

1. 01

   $\frac{d}{dx}\log x = \frac{1}{x} > 0$ and $\frac{d}{dx}x^2 = 2x > 0$ for $x > 0$: I and IV rise.

2. 02

   $\frac{d}{dx}\left(x + \frac{1}{x}\right) = 1 - \frac{1}{x^2}$ is negative on $(0, 1)$, and $\frac{d}{dx}e^{-x} = -e^{-x} < 0$.