# Critical points

1. Mar 28, 2005

### Pengwuino

Ok so i got a mathematica assignment that asks "Find and classify the critical points of $$f(x,y)= 5-10xy-4x^2+3y-y^4$$

What does this mean lol. The assignment was from 3 weeks ago and i just had spring break so my brain is completely outa wack. Does this mean classify the points as concave up and down or what? This is calculus III by the way. If someone can just lol, tell me what im being asked that would be great.

In the exercise before it we found fxx and fyy and fxy of a similar equation and then we found fxx*fyy - (fxy)^2. Im so lost lol. What is this partial derivative thing. Whats the fxx partial derivative * fxy - (fxy)^2 all about? I feel so dumb, my high school calculus teacher would be so dissappointed!

2. Mar 28, 2005

### Data

Classify the critical points as minima, maxima, or saddle points.

A partial derivative is a derivative of a function of multiple variables taken as if all but one of the variables are held constant (well, in some cases it's slightly more complicated. But just slightly!).

For a differentiable function $f(x,y)$ of two real variables with equal mixed second partial derivatives (ie. $$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$$), the quantity

$$\frac{\partial^2 f}{\partial x^2}\frac{\partial^2 f}{\partial y^2} - \left(\frac{\partial^2 f}{\partial x \partial y}\right)^2 = D$$

is called the discriminant of $f$. If $P = (x_0, y_0)$ is a critical point of $f$, then $P$ is a minimum or maximum if $D>0$ and a saddle point if $D<0$, where $D$ is evaluated at $P$. If $D=0$ then it tells you nothing. These facts can be proved using from the Taylor expansion of $f$ for analytic functions.

3. Mar 29, 2005

### dextercioby

And that "D" is actually the functions's hessian.It's a determinant

$$D=\left|\begin{array}{cc}\frac{\partial^{2}f}{\partial x\partial x}&\frac{\partial^{2}f}{\partial x\partial y}\\ \frac{\partial^{2}f}{\partial y\partial x}&\frac{\partial^{2}f}{\partial y\partial y}\end{array}\right|$$

Daniel.

4. Mar 29, 2005

### HallsofIvy

Staff Emeritus
I would say it means "open your text book and look up "critical point" (or "critical number")! It's not spring break anymore!

5. Mar 29, 2005

### dextercioby

It is to me,that's why i'm on PF...:tongue:

Daniel.