$\displaystyle z= 3x^2 + 7x + 2y^2 - 5y -xy + 13$

Find the stationary point and determine the nature of the point(g).

My question is did any step im do wrongly?

STATIONARY POINT

$\displaystyle \frac{\partial z}{\partial x} = 6x + 7 - y$

$\displaystyle \frac{\partial z}{\partial y} = 4y - 5 - x$

$\displaystyle \frac{\partial z}{\partial x} = 0 ,\frac{\partial z}{\partial y} = 0$

$\displaystyle 6x + 7 - y = 0$

$\displaystyle 4y - 5 - x = 0$

Then x and y value is:

$\displaystyle x = \frac{-3}{19}$

$\displaystyle y = \frac{23}{19}$

Substitute x and y into original equation(z), then get $\displaystyle z=\frac{3263}{361}$

stationary point is at ($\displaystyle \frac{-3}{19}$,$\displaystyle \frac{23}{19}$,$\displaystyle \frac{3263}{361}$)

Now i want find the nature of the point(g):

$\displaystyle g = (\frac{\partial^2 z}{\partial x^2})(\frac{\partial^2 z}{\partial y^2}) -(\frac{\partial^2 z}{\partial x\partial y} )^2 $

$\displaystyle = 6(4)-(-1)^2$

$\displaystyle =23>0$

Stationary point is a minimum point