Find the maximum and minimum values of the function f(x,y)

[warwick]

f(x,y) = xy -x -y -1 on the closed triangular region bounded by the x-axis and y-axis and the line x + y = 3.

I found a critical point by setting f partial dx and f partial dy each to zero.

x=1 and y=1. (1,1)

f(1,1) = 2

Beyond that, I don't know where to go with these problems. I can admit I'm almost clueless. Thanks for any help.

Would I use a Lagrange multiplier for #16?

I'm also not sure how to do the change of limits when I change to polar coordinates. I know the x = r cos theta, y = r sin theta, etc., but sometimes graphing isn't feasible to determine the theta values.

 

[Deleted member 4993]

f(x,y) = xy -x -y -1 on the closed triangular region bounded by the x-axis and y-axis and the line x + y = 3.

I found a critical point by setting f partial dx and f partial dy each to zero.

x=1 and y=1. (1,1)

f(1,1) = 2<<<< Is it? check again.

Now look at the boundary values (end-point behaviour) to check for global min/max.

Beyond that, I don't know where to go with these problems. I can admit I'm almost clueless. Thanks for any help.

Would I use a Lagrange multiplier for #16?<<<<< Yes

I'm also not sure how to do the change of limits when I change to polar coordinates. I know the x = r cos theta, y = r sin theta, etc., but sometimes graphing isn't feasible to determine the theta values.

 

[warwick]

f(x,y) = xy -x -y -1 on the closed triangular region bounded by the x-axis and y-axis and the line x + y = 3.

I found a critical point by setting f partial dx and f partial dy each to zero.

x=1 and y=1. (1,1)

f(1,1) = 2<<<< Is it? check again.

Now look at the boundary values (end-point behaviour) to check for global min/max.

Beyond that, I don't know where to go with these problems. I can admit I'm almost clueless. Thanks for any help.

Would I use a Lagrange multiplier for #16?<<<<< Yes

I'm also not sure how to do the change of limits when I change to polar coordinates. I know the x = r cos theta, y = r sin theta, etc., but sometimes graphing isn't feasible to determine the theta values.

Sorry. f(1,1) = -2 I forgot the negative. I also managed to plow my way through #16 and #18 correctly.

Generally speaking, how do I do those problems? I found the critical points of the original function, and I assume I have to find more critical points with the constraints. It's all a little fuzzy for me right now.