Question

Use the Lagrange multipliers to solve the following:

Maximize f(x,y,z) = 4x + 2y + z

subject to x² + y + z² = 1

Answer 1

Let’s define $g(x, y, z) = x^ 2 +y +z^ 2 ,$ so the problem is to find the maximum of $f(x, y, z)$ subject to the constraint $g(x, y, z) = 1.$ We have 

$$\nabla f=\lambda \nabla g$$

$$\langle4,2,1\rangle=\lambda\langle2x,1,2z\rangle$$

Reading this component by component and including the restriction we get the system of equations 

$$4=2\lambda x$$

$$2=\lambda$$

$$1=2\lambda z$$

$$x^ 2 +y+z^ 2=1$$

Then $x=1, z=\dfrac{1}{4}, \lambda=2.$

$$1+y+\dfrac{1}{16}=1$$

$$y=-\dfrac{1}{16}$$

The maximum of $f$ occurs when $x=1, y=-\dfrac{1}{16}, z=\dfrac{1}{4}.$

$$f(1,-\dfrac{1}{16},\dfrac{1}{4}) = 4(1) + 2(-\dfrac{1}{16}) + \dfrac{1}{4}=\dfrac{33}{8}$$