Let f(x)= cases ax^ 2 , & x 1, |x-1|+a, & x>1

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let f(x)={ax2,x≤1,∣x−1∣+a,x>1.f(x)=\begin{cases} ax^{2}, & x\le 1,\\ |x-1|+a, & x>1. \end{cases} Which statement about the continuity and differentiability of ff at x=1x=1 is true? (a) ff is continuous for all real aa but never differentiable at x=1x=1. (b) ff is continuous and differentiable at x=1x=1 for all real aa. (c) ff is continuous for all real aa and differentiable at x=1x=1 only when a=12a=\dfrac12. (d) ff is continuous for all real aa and differentiable at x=1x=1 only when a=1a=1.

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