How can I find the saddle point of $Z = x^2 + 3y^2 + 3xy – 6x + 3y – 6$

Question

$$Z = x^2 + 3y^2 + 3xy – 6x + 3y – 6$$

how can I find the maxima, minima and the saddle points of this equation?

I tried this by finding $fx, fy,fxx,fyy,fxy$ $$fx=2x+3y-6$$$$fy=6y+3x+3$$$$fxx=2$$$$fyy=6$$$$fxy=3$$

$$fx=fy=0$$ find $(x,y)$ $$x=15$$$$y=-8$$ But when I calculate $Z$ I get $Z=-63$ but in my textbook it says $Z=-51$ what am I doing wrong?

Answer 1

I also did the calculations, there's nothing wrong with your calculations, maybe there's a mistake in your textbook.

all your calculations are correct$$x=15$$$$y=-8$$ $$Z(15,-8)=-63$$

Answer 2

Either your textbook is incorrect or you've misread the question; I did the calculations myself and arrived at $-63$, and Wolfram Alpha agrees. Perhaps the textbook worked with a $+6$ instead of a $-6$ at the end?

Answer 3

I also got your result. And your partial derivatives look right to me. Maybe your textbook is wrong.

I would see if there are any errata available from the publisher or double-check that you have copied the function correctly

Answer 4

Your calculations are correct. I have use MatDeck to do the math and the result is the same. You can find the link for calculation in MatDeck bellow

Saddle point calculation in MatDeck