# Homework Help: Points where gradient is zero (plotting it)

1. Apr 15, 2007

### W3bbo

1. The problem statement, all variables and given/known data

A curve has equation:

x^2+2xy-3y^2+16=0

Find the co-ordinates of the points on the curve where dy/dx=0

I think I was able to differentiate it and get the coordinates fine, but I'm wanting to plot the function in Mathematica (5.2) to see if I'm right or not (BTW, I tried Ma's Dt[] and Differential[] functions, but I can't interpret the results. And plot[f, {x,-2,2}] just gives me error messages because y is undefined).

2. The attempt at a solution

x^2+2xy-3y^2+16=0

2x+2x(dy/dx)+y-3(2y(dy/dx))=0

y+2x+(dy/dx)(2x-6y)=0

(dy/dx)=-(y+2x)/(2x-6y)=0

For the fraction to equal zero, the numerator must also be zero, therefore:

-y-2x=0
y=-2x

Given this, substituting this value for y:

x^2+2x(-2x)-3(-2x)^2+16=0
x^2-4x^2-12x^2+16=0
-15x^2+16=0

x=Sqrt(960)/-30
x=Sqrt(960)/30

but it seems a little hackish to me, this from a past-paper (Edexcel Advanced Level C4, 28th June 2005), usually you get integer answers.

But besides asking if I'm right, how can I plot functions with multiple instances of x and y within? I'm guessing I'd need to convert it to a parametric somehow.

2. Apr 15, 2007

### f(x)

diff. gives $\ 2x + 2x \frac{dy}{dx} + 2y - 3(2y \frac{dy}{dx} ) = 0$

3. Apr 15, 2007

### W3bbo

Where did $+2y$ come from? I didn't have a solitary $y^2$ expression.

EDIT: Ah I see, product rule; I forgot to reapply the coefficient (2) of xy after performing the product differentiation.

Still, how can I plot the function?

Last edited: Apr 15, 2007
4. Apr 15, 2007

### Mathgician

the question is asking for the critical points of the surface right? I have a question do you need to graph this function? Do you need to find the saddle points and min max?

5. Apr 15, 2007

### W3bbo

I'm not being asked to plot the graph, and I've since found what I think are the right co-ordinates (by substituting the resolved value of y into the equation and solving) as ${ (2,0) , (-2,0) }$.

I want to plot the graph out of personal curiosity, to see what the graph actually looks like (but also to make sure I'm right). I haven't covered the plotting of implicit functions on my curriculum's syllabus though. Hence why I'm asking :)