Solve Lagrange Multipliers: Find Max/Min f(x,y)

[mirandasatterley]

Homework Statement

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

Homework Equations

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?

 

[Dick]

grad f is nonzero everywhere. So there are no critical points (local maxs or mins) inside the circle x^2+y^2=2. So the max and/or min must be ON the circle. You've found 1/(2x)=3/(2y), substitute that into x^2+y^2=2 and find the possibilities for x and y.