The abscissae of a point, tangent at which to the curve `y = e^x sin x, x in [0, pi]` has maximum slope is

Options

A. 0
B. `pi/4`
C. `pi/2`
D. `pi`

Answer

C

Explanation

The slope m of the tangent at which to the curve `y=e^x` sin x ata any pint (x,y) `m=(dy)/(dx)=e^x(sin x +cos x )`
`rArr (dm)/(dx)=2e^(cos)x and (d^2y)/(dx^2)=2e^x(cos x -sin x)`
For maximum of minimum values of m,we must have
`(dm)/(dx)=0 rArr 2e^x cos x=0 rArr x=(pi)/(2)`
Clearly `((d^2m)/(dx))_(x=(pi/2))=-2e^(pi//2) lt 0`
Hence , m is maximum when `x=(pi/2)`