Let f(x)= cases x^ 2 , & x 1, 3x-2, & x>1

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

Problem

Let f(x)={x2,x≤1,3x−2,x>1.f(x)=\begin{cases} x^{2}, & x\le 1,\\ 3x-2, & x>1. \end{cases} Determine whether f′f' exists at x=1x=1. Which statement is correct? (a) f′f' exists at x=1x=1 because the function is continuous there. (b) f′f' does not exist at x=1x=1 because the left- and right-hand slopes are different. (c) f′f' exists at x=1x=1 because both one-sided limits of the difference quotient equal 22. (d) f′f' does not exist at x=1x=1 because ff has a vertical tangent there.

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