Find a function y gx so that fx y 5x6 5x4y x2y2 y3x2 y

Find a function y = g(x) so that f(x, y) = {5x^6 - 5x^4y + x^2y^2 - y^3/x^2 - y if y notequalto x^2 g(x) if y = x^2 is continuous everywhere.

Solution

since function is continuous,
g(x) can be obtained by putting y=x^2

(5x^6 - 5x^4y +x^2y^2 -y^3)/(x^2-y)
= {5x^4 (x^2-y) + y^2 (x^2-y)}/(x^2-y)
={(5x^4+y^2)(x^2-y)/(x^2-y)}
= (5x^4+y^2)

g(x) =(5x^4+y^2) at y=x^2
g(x) = (5x^4+x^4)
= 6x^4