Mathematics

Problem

Let

f

be the function defined for all real numbers by

f(x)=x^{3}+x

. The function is differentiable and one–to–one on

\mathbb{R}

, so

f^{-1}

exists. (a) Verify that

f

is one–to–one on

\mathbb{R}

by discussing the behavior of

f'(x)

. (b) Calculate

\displaystyle f^{-1}\,'(0)