Thread: Regarding implicit differentiation

1. Regarding implicit differentiation

How do I show that the curve x^3+y^3 = 3xy is symmetrical about the line y=x??
The curve is not one-to-one so I cant show that it is a self inverse. All I can think of is to show that the expression is the same when x and y are swapped. Is that enough??

Suppose that |x| and |y| (in the above expression) are both very large. Explain why x+y~k, where k is a constant?? Because the two expressions are the same when the variables are swapped??

Thanks

2. Originally Posted by woollybull
All I can think of is to show that the expression is the same when x and y are swapped. Is that enough??

Yes, that is enough (because reflection in the line y=x is what happens when you interchange x and y).

Originally Posted by woollybull
Suppose that |x| and |y| (in the above expression) are both very large. Explain why x+y~k, where k is a constant??

Let $\displaystyle m=y/x$. Then the equation $\displaystyle x^3+y^3=3xy$ can be written $\displaystyle \frac{1+m^3}m = \frac3x$. The right-hand side of that goes to 0 as $\displaystyle |x|\to\infty$, from which you can see that $\displaystyle m\to-1$
as $\displaystyle |x|\to\infty$.

Now factorise
$\displaystyle x^3+y^3$ as $\displaystyle (x+y)(x^2-xy+y^2)$ to see that $\displaystyle x+y = \frac{3m}{m^2-m+1}\to-1$ as $\displaystyle m\to-1$.

3. Thanks a lot!!