Find the intervals in which f(x) = sin 3x, `x in [0,pi/2]` is (i) increasing, (ii) decreasing.

Options

A. `f(x)` is increasing in `[0,pi/2]`
`f(x)` is decreasing in `[pi/6,pi/2]`
B. `f(x)` is increasing in `[0,pi/6]`
`f(x)` is decreasing in `[pi/4,pi/2]`
C. `f(x)` is increasing in `[0,pi/6]`
`f(x)` is decreasing in `[pi/6,pi/2]`
D. `f(x)` is increasing in `[0,pi/6]`
`f(x)` is decreasing in `[pi/6,pi/4]`

Answer

C

Explanation

f(x) = sin 3x
` rArr f'(x) = 3 cos 3x`
(i) f(x) is increasing if `f' (x) ge 0`
`rArr 3 cos 3x ge 0`
`rArr 0 le 3x le pi/2`
`rArr 0 le x le pi/6`
`:. f(x) " is increasing in "[0,pi/6].`
(ii) f(x) is decreasing if `f'(x) le 0 `
`rArr 3 cos 3x le 0`
` rArr cos 3x le 0`
` rArr pi/2 le 3x le (3pi)/2`
`rArr pi/6 le x le pi/2`
` :. f(x) " is decreasing in "[pi/6,pi/2]`