Use the second derivative test?

Question

Let f(x, y)=x^2+y^2+xy-x+y. Use the second derivative test to show that x=1, y=-1 is a local mini-mum (and thus an absolute minimum it is the only critical point) of f. fxx=2 fyy=2 What to do next?

Answer 1

   ƒ(x, y) = x² + y² +xy - x + 1

   ƒ_x = 2x + y - 1  ,   ƒ_y = 2y + x + 1

   critical point when ƒ_x=ƒ_y=0:     (1, -1)

   ƒ_xx = 2  ,   ƒ_yy = 2  ,   ƒ_xy = 1 

The second derivative test requires you to find ƒ_xx and ƒ_xx ƒ_yy - ƒ²_xy at the critical point:

   at (1, -1):    ƒ_xx = 2   ,     ƒ_xx ƒ_yy - ƒ²_xy = (2)(2) - (1)² = 4 - 1 = 3

Since ƒ_xx>0 and ƒ_xx ƒ_yy - ƒ²_xy>0 at (1, -1), then (1, -1) is a local minimum.