ABSTRACT
We saw in Chapter 3 that the derivative of a function at a point provides a good linear approximation to the function near the point. This fact suggests that properties of the derivative at a point should be reflected in properties of the function near the point. The Inverse Function theorem and Implicit Function theorem are both instances of this phenomenon. The Inverse Function theorem states that if the derivative of a function f : V → V https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq5187.tif"/> is invertible at v → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq5188.tif"/> , then so is f, at least near v → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138695/b832c43e-68c9-4052-aad2-0d41b672a31d/content/eq5189.tif"/> . The Implicit Function theorem gives conditions (on the derivative of a function) under which a collection of equations can be solved for some of the variables in terms of the rest. We note that the Implicit Function theorem is exactly what is needed to complete the discussion of constrained maxima and minima of the previous chapter.
