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

$\frac{\pi}{2}$

The slope m of the tangent to the curve $y=e^x \sin x$ at any point (x, y) is given by

$m=\frac{d y}{d x}=e^x(\sin x+\cos x)$

$\Rightarrow \frac{d m}{d x}=2 e^x \cos x \text { and } \frac{d^2 y}{d x^2}=2 e^x(\cos x-\sin x)$

For maximum or minimum values of m, we must have

$\frac{d m}{d x}=0 \Rightarrow 2 e^x \cos x=0 \Rightarrow x=\frac{\pi}{2}$

Clearly, $\left(\frac{d^2 m}{d x^2}\right)_{x=\frac{\pi}{2}}=-2 e^{\pi / 2}<0$

Hence, m is maximum when $x=\frac{\pi}{2}$.