If the Rolle's theorem for `f(x)=e^(x)(sin x-cosx)` is verified on `[(pi)/4,(5pi)/4]` then the value of `C` is

Options

A. `(pi)/(3)`
B. `(pi)/(2)`
C. `(3pi)/(4)`
D. `pi`

Answer

B

Explanation

Given `f(x)=e^(x)(sin x-cos x)`
On differentiating both sides w.r.t.x. we get
`f'(x)=e^(x) (d)/(dx)(sin x-cos x)+(sin x-cos x)(d) (e^(x)-` (by using product rule of derrivative)
`e^(x)(cos x+sin x)+(sin x-cos x)e^(x)`
`=2e^(x)sin x`
We know that, if Rolle's theorem is verified then their exist `c in ((pi)/(4), (5pi)/(4))`, such that `f'(c)=0`
`therefore 2e^(c) sin c=0 Rightarrow sin c=0`
`Rightarrow c=(pi)/(2) in ((pi)/(4), (5pi)/(4))`