Example

Example: Let on . The minimum occurs at with . The maxima occur at and , with , . is a continuous function and so that for all in the interval . So satisfies all conditions of Theorems 1 and 2 and has therefore a unique fixed point in the interval . This fixed point is given by

> sol:=solve(x^2-4*x-1=0,x);

> evalf(sol);

> fixed_point:=sol[2];

In the interval we know there is another fixed point, but there we can not determine the necessary bound on the derivative function. This shows that the conditions in Theorems 1 and 2 are sufficient but not necessary conditions for the existence and uniqueness of fixed points.