Find the local extreme values of ƒ(x, y) = x2y on the line x + y = 3
∇ƒ = λ∇g
The Attempt at a Solution
2yxi+x^2j = λi + λj
[2yx=λ] [x^2=λ] [x+y=3]
[2yx=x^2] & [(2y)+y=3]
[2y=x] & [y=1]
This is as far as I got, and gives f(2,1)=4 as the extreme value. The solution I have (chegg.com) lists another possible solution.
On the left column, they give another solution to the equation 2y=x:
and proceed to use the "x+y=3" equation to give an alternative extrema which is f(0,3)
Can anyone explain how they made the jump from 2y=x to x=0?
Why isn't y=0 another solution? (If x equals and 2y equals x, then y must equal zero)
Thanks in advance, screenshot from solution below