The number of solutions of the equation `xe^(sinx)-cosx=0` in the interval `(0,pi//2)`, is

Options

A. 1
B. 2
C. 3
D. 4

Answer

A

Explanation

Let `f(x)=xe^(sinx)-cosx`.Then,
`f'(x)=e^(sinx)+xe^(sinx)cosx+sinx gt 0` for all `x in (0,pi//2)`
`rArr f(x)` is increasing on `(0,pi//2)`
Also, f(0)=-1 and `f((pi)/(2))=(pi)/(2)e`
So,there is a point between 0 and `pi//2`, where f(x)=0.
Hence, f(x)=0 has exactly one solution in `(0,pi//2)`