MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Let p(x)=x∣x∣ for all real numbers x.
(a) Show that p is differentiable for all x=0 and find an explicit formula for p′(x).
(b) Determine the intervals on which p is one–to–one.
(c) For each value of b listed below, state whether (p−1)′(b) exists. Where it exists, compute its value, and where it does not, provide a justification.
(i) b=−4(ii) b=0(iii) b=4