if `f(x)=(sin (x+alpha)/(sin(x+beta),alpha ne beta ` then f(x) has

Options

A. maximum at x=0
B. minimum at x=0
C. neith maximum nor minimum
D. none of these

Answer

B

Explanation

We have `f(x)=underset(cos x)overset(sinx)int(1-t+2t^3)dt`
`rArr f(x)= cos x (1-sin x+2 sin^3x)+ sin x (1- cos x + 2 cos^3 x)` [ Using Leibinitz 's Rule ]
`rArr f(x)= cos x + sin x-2 sin x cos x+2 sin x cos x(sin^2 x +cos^2 x)`
`rArr f(x)= cos x+ sin x `
For maximum of minimum we have
`f(x)=0 rArr tan x =- 1rArr x =(3pi)/(4),(7 pi)/(4)`
Now, `f(x)=- sin x+cos x `
`therefore f''((3 pi)/(4)) lt 0 and f((7 pi)/(4)) gt 0`
Hence f(x) attains local maximum at `x=(3pi)/(4)` and a minmum at `x =(7 pi)/(4)`