Let be the function defined for all real numbers by

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Let ff be the function defined for all real numbers by f(x)=x3+xf(x)=x^{3}+x. The function is differentiable and one–to–one on R\mathbb{R}, so f−1f^{-1} exists. (a) Verify that ff is one–to–one on R\mathbb{R} by discussing the behavior of f′(x)f'(x). (b) Calculate f−1 ′(0)\displaystyle f^{-1}\,'(0).

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