Lets make sure I'm doing this right

I need to find the critical points for the function

f (x, y) = e^x * sin(y)

I need to find where the slopes are horizontal or 0, so I differentiate for both x and y

I get df(x, y) for x = e^x * sin(y) and df(x, y) for y = e^x * cos(y)

For the derrivative with respect to x the only time the slope is 0 is when y=pi and for the derivative with respect to y the only time the slope is 0 is when y=pi/2. So my critical points are (pi, pi/2)

Am I right or wrong?

Also how do I tell if the critical point is a local max, local min or a saddle point?