1. The problem statement, all variables and given/known data Given f(x,y) = xy(x+2y-6), let D be the region in the plane between the hyperbola xy = 4 (let this be g) and the line x+2y-6 = 0 (let this be h). Find the maximum and minimum values of f(x,y) on D. 3. The attempt at a solution I first found the critical points for f(x,y) and it turns out that none of them are in this region. Then I used lagrange to find the critical points on the boundary: https://dl.dropbox.com/u/64325990/Photobook/Photo%202012-06-11%207%2026%2054%20PM.jpg [Broken] Solving this by hand and also confirming with wolfram I get the critical points (2,2) and (4,1). HOWEVER according to the answer key, there is one more critical point at (2√2,2/√2). They did this by solving the boundary case for xy=4 in between the regions 2 and 4 as shown here: https://dl.dropbox.com/u/64325990/Capture.PNG [Broken] I am wondering why the Lagrange method I used did not give me (2√2,2/√2) as well?