maximum and minimum values of the function f(x,y,z)=x + 2y subject to the constraints y^2 + z^2 = 289 and x + y + z = 5

Question

I know how to solve these problems with two constraints, but this one quickly falls apart and becomes realistically "unsolvable."

 

I'm trying to find the max and the min and at what points they appear.

Answer 1

Hi Kyle,

 

Let's try Lagrange Multiplier method and set g (x, y, z) = y² + z² and h(x, y, z) = x + y + z. Then your constraints are g (x, y, z) = 289, h (x, y, z) = 5.

 

Lagrange's method says that if f (x, y, z) has a min/max, you can locate them by setting the gradient of the objective function equal to an unknown linear combination of the gradients of the constraint equations. It's important to check the assumptions for this method, i) f, g, h have continuous 1st partial derivatives ii) at the extrema neither of ∇g, ∇h are 0, nor are they parallel to each other.

 

∇f = ⟨1, 2, 0⟩

∇g =⟨0, 2y, 2z⟩

∇h =⟨1, 1, 1⟩

 

Looking at the above, the assumptions hold. For unknown scalars c and d, ∇f = c∇g + d∇h. This gives you 3 equations in 5 unknowns, x, y, z, c, d, and you have two more equations from your two constraints. The first 3 equations give you

 

1 = d

2 = 2cy + d

0 = 2cz + d

 

These give y = 1/(2c) = - z so you know y = - z. 

 

Substitute into x + y + z = 5 to get x = 5.

Substitute into y² + z² = 289 to get y = ±17/√2 and z is the negative of y. Notice that you don't need to explicitly solve for c and d since you're really only interested in x, y, z.

 

Check f at the points p = (5, 17/√2, - 17/√2) and q = (5, - 17/√2, 17/√2) to see f (p) = 5 + 17√2 > 5 - 17√2 = f(q). This shows p is the maximizer and q the minimizer.