Use lagrange multipliers to find the maximum and minimum values of f subject to the given constraint, if such values exist.
f(x,y) = x+3y, x2+y2≤2
grad f = λ grad g
The Attempt at a Solution
to find critical points in the interior region,
The partial derivative of f(x,y) with respect to x is 1.
The partial derivative of f(x,y) with respect to y is 3.
g(x,y) = the constraint = x2+y2≤2, to find critical points on the boundary x2+y2=2
The partial derivative of g(x,y) with respect to x is 2x.
The partial derivative of g(x,y) with respect to y is 2y.
And normally I would set:
1 = λ 2x and 3 = λ 2y
λ = 1/2x and λ = 3/2y
so then I would set the equations equal to each other and solve the equation for x and y.
What I'm wondering though, Is whether there are actually minimum or maximum values since the partial derivatives of f(x,y) are constants?