Answer to Question #195217 in Calculus for Ichigo

Question

Using the Implicit Function Theorem, show that there exists a unique differentiable 

function g in a neighbourhood of 1 such that g (1)= 2 and F(g( y), y) = 0 in a 

neighbourhood of (1,2) where F(x,y) = x^5+y^5-16xy³-1=0.

defines the function F. Also find g′( y).

Answer 1

1 )

F(x,y) - is continuous

$\partial_y F=5y^4-48xy^2<0\\ \partial_x F=5x^4-16y^3>0$

F(x,y)

is monotone in the vicinity of this point

$g'(y)=-\frac{5y^4-48xy^2}{5x^4-16y^3};g'(1)=-\frac{-91}{64}=\frac{91}{64}$