Prove that there are no stationary points for x^3 + xy^2 - y^3 = 5?

Question

Prove that there are no stationary points for x^3 + xy^2 - y^3 = 5?

Answer 1

A point is a stationary point if dy/dx = 0.

 

x³ + xy² - y³ = 5

 

3x² + y² + 2xy(dy/dx) - 3y²(dy/dx) = 0

 

(dy/dx)(2xy - 3y²) = -3x² - y²

 

dy/dx = (-3x² - y²) / (2xy - 3y²)

 

If dy/dx = 0, then -3x² - y² = 0

 

-3x² - y² = 0 only if x = y = 0.

 

But, (0,0) does not lie on the graph of x³ + xy² - y³ = 5.

 

So, there are no stationary points for the given curve.