Let be differentiable on an open interval containing with and
MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Let f be differentiable on an open interval containing x=2 with f(2)=5 and f′(2)=0. Which statement best explains why the inverse function of f is not differentiable at x=5?
(a) Because f′(2)=0, the inverse has derivative 0 at x=5.
(b) Differentiability of f at x=2 guarantees differentiability of f−1 at x=5 regardless of the value of f′(2).
(c) Because f′(2)=0, the reciprocal needed to compute (f−1)′(5) is undefined, so the inverse is not differentiable there.
(d) Since f′(2)=0, (f−1)′(5)=01=0, so the inverse is differentiable at x=5.