**Exercise 47**
If

*f* is continuously differentiable and

*f'*(

*x*_{0})

0 then show
that

*f* is monotonic in some interval around

*x*_{0}. Hence show that

*f* has a inverse

*g* (as in the exercise above) in some small enough
interval around

*f* (

*x*_{0}).

**Exercise 48**
If

*f* can be expressed as

*f* (*x*) = *f* (*x*_{0}) + *f*_{1}(*x* - *x*_{0}) + ... + *f*_{n}(*x* - *x*_{0})^{n} + *o*((*x* - *x*_{0})^{n})

with

*f*_{1} 0, then show that the inverse function

*g*(

*y*) has the
following form where

*y*_{0} =

*f* (

*x*_{0}).

*g*(

*y*) =

*x*_{0} +

(

*y* -

*y*_{0}) -

(

*y* -

*y*_{0})

^{2} + ... +

*g*_{n}(

*y* -

*y*_{0})

^{n} +

*o*((

*y* -

*y*_{0})

^{n})

where

*g*_{n} is of the form

*P*_{n}(

*f*_{1},...,

*f*_{n})/

*f*_{1}^{n + 1}, where

*P*_{n} is a polynomial function.