Consider the function f R2 rightarrow R such that fx y y2

Consider the function f: R^2 rightarrow R such that f(x, y) = y^2 - x^2 (1 - x) and let C subset R^2 be the curve defined by f^-1 (0) (you may find it helpful to make a sketch of this curve). Show that C may be represented either in the form y = phi (x) or x = psi (y) near any (x, y) elementof C\\{(0, 0)}. Write down the formula for the implicit function to be used near (1, 0).

Solution

Given that ----- f(x,y) = y² - x²(1-x)

We have to find function at (1,0).

So put y = 0 and x = 1.

f(1,0) = (0)² - (1)²(1-1)

f(1,0) = 0 - 1(0) = 0