Chapter 3, Problem 4RWDT

To determine

To show:. Give an example of a function which is one-one, but whose inverse fails to be differentiable at a point.

Explanation of Solution

Given information :

The function is differentiable.

Formula used:

To show that a function $f\left(x\right)$ is injective, assume two points $a\text{and}b$ with $f\left(a\right)=f\left(b\right)$ and then show that $a=b$

Proof:

Assume that the function is $f\left(x\right)=x^3$ .

The graph of the function is below:

  

Graphically the function is one-one as horizontal line cuts the curve at most one.

The derivative is as follow:

  $\begin{array}{c}\frac{d}{dx}\left[f\left(x\right)\right]=\frac{d}{dx}\left[x^3\right]\\ =3x^2\end{array}$

which is zero for $x=0$ .

Move the positive part to the right.

The graph of the inverse function is $\sqrt[3]{x}$

  

The inverse function will have a jump.

This is because any place where differentiable function has zero slope, the inverse function has infinite slope.