Give an example of a function fRR that is differentiable and

Give an example of a function f:R->R that is differentiable and one to one but f\'(x) = 0 for some x in the reals (explain clearly)

Solution

f(x)=x^3

f\'(x)=3x^2

f\'(0)=0

Let, f(x)=f(y)

x^3=y^3

x^3-y^3=(x-y)x^2+y^2+xy)=0

x-y=0 , x^2+y^2+xy=0

(x/y)^2+(x/y)+1=0

This has no real solutions

So, x=y

Hence, f is one to one, differentiable and f\'(x)=0 for x=0