Billy has 1 gallon of paint. He is going to pour it into a paint tray that measures 10 inches wide, 14 inches long, and 4 cm dee
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The Lagrangian is
with partial derivatives (set equal to 0)
Case 1: If , then
Then
So we have two critical points, and
Case 2: If , then in the first equation we get
and from the third equation,
Then
but there are no real solutions for , so this case yields no additional critical points.
So at the two critical points we've found, we get extreme values of
(min)
and
(max)