Partial Derivatives

I have the following equation:

z(x,y)=3x^3-5y^2-225x+70y+23

I have solved for x=+5 x=-5

solving for y=7 but putting them into the original function z(x,y)=/=0

So, trying to input the values obtained for x and solving the resulting quadratic formula give imaginary roots.

Have I done something wrong here? I'm just trying to find the critical point for the function when it is optimised.

Any help would be great!

Thanks

For critical point solve simultaneously
$\partial \frac{dz}{dx}=0$ and


The solution for x is other than you wrote.
FOr type of critical point you should to give the values of
$z''_{xx}=\partial{d^2z}{dx^2}(x_0,y_0)$
$z''_{yy}=\partial{d^2z}{dy^2}(x_0,y_0)$
$z''_{xy}=\partial{d^2x}{dxdy}(x_0,y_0)$
and examine sign of $z''_{xx}\cdot z''_{yy}-(z''_{xy})^2$

But how can you solve these simultaneously?

Having one equal to the other?




$\partial \frac{dz}{dx}=9x^2-225$

But how can you solve these simultaneously?

Having one equal to the other?




$\partial \frac{dz}{dx}=9x^2-225$