# Need help with Lagrange multiplier question?

Apr 15, 2018

See below.

#### Explanation:

$L \left(x , y , \lambda\right) = 5 {x}^{2} + 6 {y}^{2} - x y + \lambda \left(x + 2 y - 24\right)$

$\nabla L = \left(\begin{matrix}10 x - y + \lambda \\ - x + 12 y + 2 \lambda \\ x + 2 y - 24\end{matrix}\right) = \left(\begin{matrix}0 \\ 0 \\ 0\end{matrix}\right)$

Now solving for $x , y , \lambda$

$x = 6 , y = 9 , \lambda = - 51$ ----(*)

The qualification is done over $f \circ g$ or

$f \circ g = 7 {x}^{2} - 84 x + 864$

hence $x = 6 , y = 9$ as determined in (*) is a local minimum.