(adsbygoogle = window.adsbygoogle || []).push({}); 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

x^{3}+y^{3}=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, F_{y}, doesNOTequal 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 x^{3}+y^{3}-6x-2y=0=F(x,y)

Then I use the formula:

y'(x)=(-F_{x})/(F_{y})

Thus y'(x)=(-3x^{2}+6)/(3y^{2}-2)

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

(F_{y}) doesNOTequal 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 IONLYhave to show that that y can be

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

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

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