Let `f(x) = 2sin^3x-3sin^2+12sinx+5, 0lexlepi/2` Then f(x)is

Options

A. decreasing on `[0,pi//2]`
B. increasing on `[0,pi//2]`
C. increasing on `(0,pi//4)` and decreasing on `(pi//4,pi//2)`
D. none of these

Answer

B

Explanation

We have
`f(x) = 2sin^3x-3sin^2+12sinx+5,`
`rArr f'(x) = 6sin^2xcosx-6sinxcosx+12cosx`
`rArr f'(x) = 6(sin^2x-sinx+2)cosx`
Since discriminant of the quadratic `sin^2xsinx+2` in sinx is negative
`therefore sin^2x-sinx+2gt0 " for all " x in R`
`rArr` Sign of f'(x) is same as that of cosx
` rArr f'(x) gt0 " for all " x in (0,pi//2)`
` rArr` f(x) is increasing on `[0,pi//2]`