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).

ganolrifv9 2022-07-29 Answered
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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Reinfarktq6
Answered 2022-07-30 Author has 18 answers
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)=0
Here 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)|<1} in which F / y 0
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