Which of the following functions are strictly decreasing on `[0,pi/2]`(A) `cos x` (B)`cos 2x` (C) `cos 3x` (D) `t a n x`

Explanation

(a) `f(x) = cos x`
`rArr f'(x) =- sin x`
If `0 lt x lt pi/2`
then ` 0 lt sin x lt 1`
`rArr -1 lt - sin x lt 0`
`rArr -1 lt f'(x) lt 0`
`:. ` f(x) = cos x is strictly decreasing in `(0, pi/2)` .
(b) `f(x) = cos 2x`
` rArr f'(x) =- 2 sin 2x`
If ` 0 lt x lt pi/2`
then ` 0 lt 2x lt pi`
` rArr 0 lt sin 2x lt 1`
` rArr -2 lt - 2 sin 2x lt 0`
` :. f(x) = cos 2x" is strictly decreasing in "(0,pi/2)`.
(c) f(x) = cos 3x
`rArr f'(x) = - 3 sin 3x`
Let `f'(x) = 0`
`rArr sin 3x = 0`
`rArr 3x = pi`
` rArr x = pi/3`
Divide the interval `(0,pi/2)"into "(0,pi/3) cup )pi/3,pi/2)`.
If ` x in (0, pi/3)" then "sin 3x gt 0`
`rArr -sin 3x lt 0`
` rArr f'(x) lt 0`
`rArr f(x)` = cos 3x is strictly decreasing in `(0, pi/3)`.
If ` x in (pi/3,pi/2)" then " sin 3x lt 0`
` rArr - sin 3x gt 0`
` rArr f'(x) gt 0`
`rArr f(x) = cos 3x" is strictly increasing in "(pi/3, pi/2)`.
`:. f(x) = cos 3x" is not strictly increasing in "(0,pi/2)`.
(d) f(x) = tan x
` rArr f'(x) = sec^(2) x gt 0" if " x in (0,pi/2)`
`:. f(x)= ` tan x is not strictly decreasing in `(0,pi/2)`.