If we have a derivative, such as $(e^x-1)/x$, could we test the value of $x$ at $0$ to see it it’s a relative minimum or maximum (using the first derivative test) at that point? In other words, is it possible to use values that produce indeterminate form as candidates for the first derivative test?
2026-03-26 20:39:36.1774557576
Can indeterminate form be used as a critical point?
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The first derivative test is concerned with critical points $c$, where $f$ is differentiable near to $c$ (but not necessarily including $c$).
Critical points are not just those where $f'(c) = 0$, but where $f'(c)$ does not exist.
In particular, the first derivative test is concerned with whether $f'$ changes sign at $c$, e.g. $f'(x)<0$ for $x$ just below $c$ and $f'(x) > 0$ for $x$ just above $c$, or the reverse.
The value of $f'(c)$ itself (or even the existence of it) is not a concern for the first derivative test. Hence, you can use the first derivative test on $f(x)=|x|$ at $x=0$, as an example.