Show that y can be written as a function of x

[number0]

Homework Statement

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.

Homework Equations

Implicit Function Theorem

The Attempt at a Solution

By the Implicit Function Theorem, if the partial derivative of y, F_y, 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)=(-F_x)/(F_y)

Thus y'(x)=(-3x²+6)/(3y²-2)

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

(F_y) does NOT equal zero.

At (0,0), F_y=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.

 

[Coto]

This looks good.