Using Kuhn-Tucker condition, solve the following NLPP.

Maximise, $z=2x_{1}^{2}-7x_{2}^{2}+12x_{1}x_{2}$

Subject to, $2x_{1}+5x_{2} \le 98$

$x_{1}, \ x_{2} \ge 0$

Solution:

$z=2x_{1}^{2}-7x_{2}^{2}+12x_{1}x_{2}$

Subject to, $2x_{1}+5x_{2} \le 98$

$x_{1}, \ x_{2} \ge 0$

We write the given problem as

$f(x_{1}, \ x_{2})=2x_{1}^{2}-7x_{2}^{2}+12x_{1}x_{2}$ -------------(1)

$h(x_{1}, \ x_{2})=2x_{1}+5x_{2}-98$ ----------------(2)

Now, Kuhn-Tucker condition are,

$\cfrac{\delta f}{\delta x_{1}}-\lambda \cfrac{\delta h}{\delta x_{1}}= 0$ -------------(3)

$\cfrac{\delta f}{\delta x_{2}}-\lambda \cfrac{\delta h}{\delta x_{2}}= 0$ -------------(4)

$\lambda h(x_{1}, \ x_{2})=0, …