The implicit function theorem gives conditions under the relationship G(x,y)=0 defines y implicitly as a function of x. Use the implicit function theorem to show that the relationship x+y+e^{xy}=0 defines y implicitly as a function of xnear the point (0,-1).

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Let F(x,y,) be a continuous function with continuous partial derivative defined on an open set S containing the point   P  =  (      x          0        ,      y          0        ). If   ∂  F      /    ∂  y  ≠  0 at P then there exists a region R about       x          0       such that for any x in R there is a unique y such that F(x,y)=0Here   F  (  x  ,  y  )  =  x  +  y  +      e          x      y       is continuous andat the point P(0.-1) the partial derivative   ∂  F      /    ∂  y  =  1  +  x      e          x      y       at (0,-1) is nonzero.Hence from the Implicit function theorem, we can say that ,there exists a function in some neighbouhoodof (0,-1) such that y=f(x).One possible entry for neighbourhood is{(x,y):|(x,y)-(0,-1)|&amp;lt;1} in which   ∂  F      /    ∂  y  ≠  0

The implicit function theorem gives conditions under the relationship G(x,y)=0 defines y implicitly as a function of x. Use the implicit function theorem to show that the relationship x+y+e^{xy}=0 defines y implicitly as a function of xnear the point (0,-1).

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