First derivative test and one-sided derivative test are incomparable

From Calculus
Jump to: navigation, search

The first derivative test and one-sided derivative test are incomparable: it is possible to have situations where either test is conclusive and the other isn't.

A full table of possibilities is below.

Example function Conclusion Is the first derivative test conclusive? Is direct analysis of one-sided derivatives conclusive?
f(x) := x^2, c = 0 local minimum Yes No, because both one-sided derivatives are zero
f(x) := |x|, c = 0 local minimum Yes Yes
f(x) := \left\lbrace \begin{array}{rl} |x|(2 + x \sin(1/x)) & x\ne 0 \\ 0, & x = 0 \\\end{array}\right., c = 0 local minimum No Yes
f(x) := \left\lbrace \begin{array}{rl} |x|(2 + \sin(1/x)) & x\ne 0 \\ 0, & x = 0 \\\end{array}\right., c = 0 local minimum No No