Calc. 3 max and min of a region

[bassprosox]

having difficulty getting started on this problem, searched through Calculus Early Transcendentals by Rogawski but cant seem to get any leads, any help would be greatly appreciated

a. Find the absolute maximum and minimum of the function g ( x , y)= (x^3)?2xy+(y^2)?y
on the triangular region R with vertices (-1,0), (1,0), (1,2).
b. What is the range of g(x,y) if the domain of this function is restricted to the region R?
c. What is the range of the function if its domain is restricted to the interior of R?

Thanks, Travis

 

[akoaysigod]

It helps to draw a picture of the region which you've probably already have done.

The first thing you need to do is find the critical points of g(x,y) to do this find the partial derivatives with respect to x and y and find out where they equal 0.

gx = 3x^2 - 2y = 0
and
gy = -2x+2y^2 -1 = 0

unless I made a mistake.
If there are any critical points inside the region you'll have to perform the second derivative test to determine whether or not the points mean anything, if they're outside the bounds then it doesn't even matter. The next part is probably where you're having trouble. You have to determine if the max or min occur on the boundaries. Its easiest to do by getting one variable in your function and then its like a calc I problem.

One boundary (L1) is the x-axis which is y=0. So you're function would be g(x,0).
L2 could be x = 1. g(1, y)
L3 is a bit strange. Its the line where y = x + 1. g(x, x+1)

Then you test these points between the region boundaries? I don't know how to say that basically -
if you have g(x, 0) then x varies between -1 =< x =< 1 and test it, etc.

As far as the ranges go I'm not sure. Someone else can probably give you more insight into that but this should at least get you started.