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.