# Thread: Maximizing Linear Function

1. ## Maximizing Linear Function

If we have

$\displaystyle f(x,y,z) = 3x + 2y +5z$

Subject to the following constraints:

$\displaystyle g(x,y,z) = x + 2y+z \leq 430$
$\displaystyle h(x,y,z) = 3x+2z \leq 460$
$\displaystyle i(x,y,z) = x+4y \leq 420$

and we want to maximize (or minimize) it, how would we solve this? My lecture is originally about the Simplex method. But I was wondering, could we use Lagrange method here? I tried it but after finding partial derivatives of f(x,y,z) and the constraints I only come up with a solution to the scalar multiples of the constraints and I don't know what to do after that:

$\displaystyle \nabla f(x,y,z) =\lambda \cdot g(x,y,z) + \theta \cdot h(x,y,z) + \eta \cdot i(x,y,z)$

I get:

$\displaystyle \lambda = 5 , \theta = 0 , \eta = -2$

2. ## Re: Maximizing Linear Function

The Lagrange method is for nonlinear optimisation, as it involves finding stationary points. Linear functions do not have these. You will need to use the Simplex Method.

3. ## Re: Maximizing Linear Function

thank you Prove It