First derivative test and one-sided derivative test are incomparable

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
$f(x):=\left\lbrace {\begin{array}{rl}|x|(2+x\sin(1/x))&x\neq 0\\0,&x=0\\\end{array}}\right.$, $c=0$	local minimum	No	Yes
$f(x):=\left\lbrace {\begin{array}{rl}|x|(2+\sin(1/x))&x\neq 0\\0,&x=0\\\end{array}}\right.$, $c=0$	local minimum	No	No