1. ## Critical Points

So I have $\displaystyle f(x,y)=x^3y+12x^2-8y$ and I need to find the maxima and minima. I calculated that $\displaystyle \bigtriangledown f = <3yx^2+24x,x^3-8>$, and by setting both to zero I figured out that there are critical points where $\displaystyle x = 0, x = -8/y, x=+-2$. Here's where I'm running into problems. I know how to find the x values, but I'm confused about how to find the corresponding y values for these critical points. For the first one, clearly y=0 when x = 0, but I'm not sure of the others because plugging them in to the 2 equations doesn't give me anything definite. In cases like this, what is the best way to find the corresponding y values of the critical points?

2. Hello, Chaobunny!

$\displaystyle f(x,y)\:=\:x^3y+12x^2-8y.\;\;\text{ Find the maxima and minima.}$

$\displaystyle \text{I found that }\nabla f \:=\:\langle 3yx^2+24x,\,x^3-8\rangle$

$\displaystyle \text{Setting both to zero, I found that there are critical points}$
. . $\displaystyle \text{where: }\:x = 0,\:x = -\frac{8}{y},\:x= \pm 2$ . . How?

We have: .$\displaystyle \begin{Bmatrix}f_x &=& 3x^2y + 24x &=&0 & [1] \\ f_y &=& x^3-8 &=& 0 & [2] \end{Bmatrix}$

From [2], we have: .$\displaystyle x^3 \:=\:8 \quad\Rightarrow\quad x \;=\:2$

Substitute into [1]: .$\displaystyle 12y + 48 \:=\:0 \quad\Rightarrow\quad y \:=\:\text{-}4$

We have one critical point: .$\displaystyle (2,\,\text}-}4,\,48)$

We have: .$\displaystyle \begin{Bmatrix}f_{xx} &=& 6xy + 24 \\ f_{yy} &=& 0 \\ f_{xy} &=& 3x^2 \end{Bmatrix}$

$\displaystyle D \;=\;(f_{xx})(f_{yy}) - (f_{xy})^2 \;=\;(6xy+24)(0) - (3x^2)^2 \;=\;-9x^4$

At $\displaystyle (2, \text{-}4, 48)$, we have: .$\displaystyle D \:=\:-9(2^4) \:=\:-144$

Since $\displaystyle \,D$ is negative, there is a saddle point at $\displaystyle (2, \text{-}4, 48).$

3. Thank you very much! So, as a clarification, if there isn't an obvious corresponding y value, then is there no critical point at that location?

4. I have no idea what you mean by "an obvious corresponding y value". The point is that both equations, $\displaystyle 3yx^2+ 24x= 0$ and $\displaystyle x^3- 8= 0$ must be true at a critical point. The only solution to $\displaystyle x^3- 8= 0$ is x= 2 (NOT x= -2. If x= -2, $\displaystyle x^3- 8= -8- 8= -16$, not 0) and, putting that back into the first equation, $\displaystyle 12y+ 48= 0$ so that $\displaystyle y= -4$.

The fact that x= 0 satisfies the first equation is irrelevant- it does not satisfy the second equation for any value of y.