00:01
Hi, in this video we're given the function f of x y is equal to 2 x squared plus 2 y squared plus 5 x plus 5 y minus 2, and we're looking at the domain x squared plus y squared is less than or equal to 36.
00:23
And we want to find the absolute extrema, so the first thing we're going to do is we're going to set, we're going to calculate the partial derivatives and set them equal to so we get 4 x plus 5 equals 0, and then 4 y plus 5 equals 0.
00:45
So the only critical point we get is going to be negative 5 fourths negative 5 fourths.
00:54
So this is the only point we have to care about checking, otherwise the extrema are going to be on the boundary.
01:04
And so on the boundary, and by boundary i mean on the circle, so we have our circle which has radius 6, and then we have our point negative 5 4 negative 5 4 somewhere over here.
01:34
And so on the boundary we know that f of x y is going to be 2 times x squared plus y squared plus 5 times x plus y minus 2.
01:46
And since x squared plus y squared is 36, this is just 72 plus 5 times x plus y minus 2, 70 plus 5 times x plus y.
02:02
And now the 70 is a constant, so all we care about is the 5 times x plus y.
02:07
And we know x plus y is going to be maximized when x and y are both as big as possible...