# Show that y can be written as a function of x

1. Jun 6, 2010

### number0

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

Show that y can be written as a function of x near the point (x,y)=(0,0) with

x3+y3=6x+2y and what y'(x) is equal to.

2. Relevant equations

Implicit Function Theorem

3. The attempt at a solution

By the Implicit Function Theorem, if the partial derivative of y, Fy, does NOT equal 0 at a certain point then y can be

written as a function of x at that certain point.

First, I set the equation to x3+y3-6x-2y=0=F(x,y)

Then I use the formula:

y'(x)=(-Fx)/(Fy)

Thus y'(x)=(-3x2+6)/(3y2-2)

In other words, I am trying to show that the partial derivatives with respect to y

(Fy) does NOT equal zero.

At (0,0), Fy=3(0)2-2= -2

And at y'(0,0)=-3

I am not sure if I applied the Implicit Function Theorem correctly in this problem. Can

anyone see if I miss anything? Remember I ONLY have to show that that y can be

written as a function of x near the point (x,y)=(0,0), nothing less or more.

Last edited: Jun 6, 2010
2. Jun 7, 2010

### Coto

This looks good.