Let `f(x)=ab sin x+bsqrt(1-a^(2))cosx+c`, where `|a|lt1,bgt0` then

Options

A. maximum value of `f(x)=-cos^(-1)a`
B. `f(x)=c` if `x=cos^(-1)alpha`
C. `f(x)=c` if `x=-cos^(-1)alpha`
D. `f(x)=c` if `x=cos^(-1)alpha`.

Answer

A, B, C

Explanation

`f(x) =ab sin x + bsqrt( 1- a^(2)) cos x + c`, where `|a| lt 1, b lt 0`
`f(x) = sqrt(a^(2)b^(2)+ b^(2) - b^(2)c^(2)) sin (x + alpha ) + c`
`" " = bsin (x + alpha) + c` , where `tan alpha = (bsqrt(1-a^(2)))/(ab) = (sqrt(1-a^(2)))/(a)`
`" " = b cos(x - alpha) +c`, where ` tan alpha = (ab)/(bsqrt(1-a^(2))) = (a)/(sqrt(1-a^(2)))`
`f(x)_(max) - f(x)_(min) = c + b - (c-b) = 2b`
`f(x) = c " if " x + alpha =0`
or `x = - alpha `
or ` x = - cos^(-1)alpha`