Implicit function theorem for $x^2y^3+x^3z+2yz^4=4$ at $(1,1,1)$

The point $P = ( 1 , 1 , 1 )$ lies on the surface S in $\mathbf { R } ^ { 3 }$ defined by

\[ x ^ { 2 } y ^ { 3 } + x ^ { 3 } z + 2 y z ^ { 4 } = 4 \]

Prove that there exists a differentiable function $f ( x , y )$ defined in an open neighborhood N of $( 1 , 1 )$ in $\mathbf { R } ^ { 2 }$ such that $f ( 1 , 1 ) = 1$ and $( x , y , f ( x , y ) )$ lies in S for all $( x , y ) \in \mathcal { N }$