Consider . Which statement correctly describes the behavior of at…
MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider h(x)=3x+∣x∣. Which statement correctly describes the behavior of h at x=0?
(a) h is continuous at x=0, but h′(x) approaches +∞ from the right and −∞ from the left, so the graph has a cusp at the origin.
(b) h is continuous at x=0 and has a corner there because the ∣x∣ term yields slopes +1 and −1.
(c) h is continuous at x=0, but not differentiable; both one-sided derivatives grow without bound to +∞, giving a vertical tangent at the origin.
(d) h is not continuous at x=0 because ∣x∣ is not differentiable there.