- #1
cdotter
- 305
- 0
Homework Statement
f(x,y)=xy-y^2 is bound by a rectangle with vertices at (2,-1), (2,2), (-1,2), and (-1,-1.)
Find all 2D, 1D, and 0D critical points.
Homework Equations
The Attempt at a Solution
2D Points:
[tex]f(x,y)=xy-y^2[/tex]
[tex]f_x=y=0 \Rightarrow y=0[/tex]
[tex]f_y=x-2y=0 \rightarrow x-2(0)=0 \Rightarrow x=0[/tex]
Therefore the critical point is (0,0)
1D Points:
The domain is a rectangle. The edges are
Top: (x,2) [itex]\rightarrow (x)(2)-2^2=2x-4; \frac{d}{dx}=2[/itex]
Bottom: (x,-1) [itex]\rightarrow(x)(-1)-(-1)^2=-x-1; \frac{d}{dx}=-1[/itex]
Left: (-1,y) [itex]\rightarrow(-1)(y)-y^2=-x-1; \frac{d}{dy}=-y-y^2=0 \Rightarrow y=-\frac{1}{2}[/itex]
Right: (2,y) [itex]\rightarrow(2)(y)-y^2=2y-y^2; \frac{d}{dy}=2-2y=0 \Rightarrow y=1[/itex]
That gives me two 1D points: (-1, -1/2) from the left edge and (2,1) from the right edge. My professor says the left edge is (-1/2, 2.) What am I doing wrong? That doesn't even make sense because 2 is a y coordinate which would be a top or bottom edge.
0D Points: The rectangle vertices.