Discuss the nature (saddle point, extremal point) of the critical points $p_o$ of the function described by:
$f(x,y) = 4x^2 -12xy + 9y^2$.
Then,
(1) $f_1 (x,y) = 8x - 12y$
and
(2) $f_2 (x,y) -12x + 18y$.
To get the critical points of the function $f(x,y)$, I set (1) and (2) equal to $0$ and solve the simultaneous equations. So,
$8x - 12y = 0$
$-12x + 18y = 0$.
Since These two equations are scalar multiples of each other, I get that $t(\frac{3}{2},1)$ is the set of critical points for the function.
Now, $f_{12} = -12$, $f_{11} = 8$, and $f_{22} = 18$. From my textbook, $\Delta = (f_{12} (p_o))^2 - f_{11} (p_o)f_{22} (p_o)$, which means that $\Delta = 144 -(18)(8) = 0$. My textbook says that if $\Delta = 0$, then the nature of $p_o$ can't be determined by the $\Delta$ formula. Are there other ways to figure out the nature of the critical points of $f$?
$$(4x^2-12xy + 9y^2)=(2x-3y)^2\ge 0$$
Hence it attains minimum value when $2x=3y$.