For , which of the following statements is most accurate regarding the…
MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
For q(x)=(x−3)1/3+(x−3)2, which of the following statements is most accurate regarding the differentiability of q at x=3?
(a) q is differentiable at x=3 with q′(3)=0 because the quadratic term dominates and smooths the graph near x=3.
(b) q is differentiable at x=3 because both individual terms are differentiable at that point, so their sum must be.
(c) q is not differentiable at x=3 since the left-hand derivative approaches −∞ while the right-hand derivative approaches +∞.
(d) q is not differentiable at x=3 because the (x−3)1/3 term produces an infinite derivative, and the quadratic term does not remove this behavior.