# How do you find the maximum value of the function #f(x,y,z)= x+2y-3z# subject to the constraint #z=4x^2+y^2#?

##### 2 Answers

#### Explanation:

Forming the lagrangian

with

The stationary points are computed solving

or

and solving for

and the maximum value is

NOTE: To qualify the stationary point it is necessary to form

and then calculate

As we can observe,

Use a Lagrange Multiplier .

#### Explanation:

Given:

The Lagrange function is:

We compute the partial derivatives:

Set these 4 derivatives equal to zero and then solve them as a system of equation:

Solve equation [3] for

Substitute -3 for

Substitute -3 for

Use equation [4] to find the value of z:

Note: One cannot use the second derivative to test whether the Lagrange multiplier has given you a maximum or a minimum; the only way to determine whether the value is a local maximum is perturbation of values. I will leave that exercise to you.