Consider the derivative f'(x)= cases - ,|x+2|^ -1/2 , & x-2 cases and suppose is…
MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Consider the derivative
f′(x)={−∣x+2∣−1/2,∣x+2∣−1/2,x<−2x>−2
and suppose f is continuous for all real x. Which statement best classifies the behavior of f at x=−2?
(a) f has a vertical tangent line at x=−2 because the slopes approach +∞ from both sides.
(b) f is differentiable at x=−2 since f is continuous and f′(x) is given piece-wise.
(c) f has a cusp at x=−2 because the slopes approach −∞ from the left and +∞ from the right.
(d) f has a corner at x=−2 because the one-sided slopes are finite but unequal.