The range of the functions `f : [0,1] to R`, given by `f(x) = x^(3) - x^(2) + 4x + 2 sin^(-1)x` , is

Options

A. ` [-pi-2, 0] `
B. `[2, 3 ] `
C. `[ 0, 4 + pi]`
D. `[0, 2+ pi]`

Answer

C

Explanation

We have, `f(x) = x^(3) - x^(20 + 4x + 2 sin ^(1) x`
`rArr f(x) = 3x^(2) - 2x + 4 +(2)/(sqrt(1=x^(2)))`
`rArr f(x) lt 0 ` for all `x in [0,1]" "[because 3x^(2) -2x+ 4 gt 0"for al" x in R]`
`rArr f(x)` is increasing function on `[0,1]`
Hence,range`(f) =[(f (0),f(1)]=[0,4 + pi]`