Question

Find the absolute maximum and minimum values for f(x,y)=xy on the rectangle R defined by -1 13 years ago

Answer 1

Step #1: find the critical point of f.
f_x = y
f_y = x
f_x = f_y = 0 -> x = y = 0
therefore (0,0,0) is our critical point.

Answer 2

Step #2: find critical points along the boundaries.
horizontal:
g = f(-1,y) = -y
g' = 0 -> y = 0
f(-1,0)=0
so (-1,0,0) is a critical point.
h = f(1, y) =y
h' = 0 -> y = 0
f(1,0)=0
so (1,0,0) is a critical point.
vertical:
g = f(x, -1) = -x
g' = 0 -> x = 0
f(0, -1) = 0
so (0,-1,0) is a critical point.
h = f(x, 1) = x
h' = 0 -> x = 0
f(0,1) = 0
so (0,1,0) is a critical point.

Answer 3

Step #3: determine the value of the function at the boundary corners.
f(-1,-1)=1
f(-1,1)=-1
f(1,-1)=-1
f(1,1)=1
so our corner points are (-1,-1,1), (-1,1,-1), (1,-1,-1), (1,1,1).

Answer 4

Step #4: conclude which constitute the absolute extrema...
(-1,1,-1) and (1,-1,-1) are the absolute minima while (-1,-1,1) and (1,1,1) are the absolute maxima.

Answer 5

Thanks for your help! :D