Which of these functions are strictly decreasing on $\left(0, \frac{\pi}{3}\right)$?

I. $\cos x$

II. $\cos 2x$

III. $\cos 3x$

IV. $\tan x$

Explanation

1. 01

   $\cos kx$ falls while $kx$ runs from 0 to $\pi$. On $\left(0, \frac{\pi}{3}\right)$, $3x$ only reaches $\pi$, so $\cos x$, $\cos 2x$ and $\cos 3x$ all fall.

2. 02

   $\tan x$ has derivative $\sec^2 x > 0$, so it rises.