Use Lagrange multipliers to find the maximum and minimum values of f(x,y)=2x−5y subject to the constraint x2+3y2=111, if such values exist.

Question

Use Lagrange multipliers to find the maximum and minimum values of  subject to the constraint , if such values exist.

Answer 1

We need to find the min and max values of f(x,y)=2x-5y subject to the constraint g(x,y)=x²+3y²=111.

Using Lagrange multipliers, we solve the equations ∇f= λ∇g and g(x,y)=111. This gives

2=λ2x (1)

-5=λ6y (2)

From (1) we have λ=1/x. Substituting this into (2)

we have x=-6y/5 (3).

Substituting (3) into the constraint g(x,y) we have

(-6y/5)²+3y²=111⇒y²(-36/25 + 3)=111⇒y=±5√37/√13.

We have corresponding points (-6√37/√13, 5√37/√13) and (6√37/√13, -5√37/√13).

Evaluating f at these critical points:

f(-6√37/√13, 5√37/√13)=13√37/√13 ≈ -62.42 (abs min)

f(6√37/√13, -5√37/√13) 37√37/√13 ≈ 62.42 (abs max)