Suppose f(x,y)=xy−ax−by: How many local minimum, maximum and saddle points?

Since f_x = y - a and f_y = x - b, setting these equal to 0 yields the critical point
at (x, y) = (b, a).

Classifying this point:
f_xx = 0, f_yy = 0, f_xy = 1.
==> D = (f_xx)(f_yy) - (f_xy)^2 = 0 - 1^2 = -1 < 0.

Hence, we have a saddle point at (x, y) = (b, a).
(Moreover, there are neither local maxima nor local minima.)

I hope this helps!