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We know that $x=\pm 1$ are in the domain of $f(x)=\sin^{-1}x$ but not in the domain of $f'(x)$. Can we say that they are critical points?

Umesh shankar
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2 Answers2

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I agree with subrosar, however, if the definition is for the purpose of analyzing global or local maximums and minimums of a function, it is simply cleaner to take this sort of definition like off Wikipedia (of course many analysis texts give this as well):

"A critical point of a function of a single real variable, $f(x)$, is a value $x_0$ in the domain of $f$ where it is not differentiable or its derivative is $0$ ($f ′(x_0) = 0)$."

The catch-all is nice, as you no longer have to stress "check the endpoints" when looking for extrema; the endpoints are always (see comments below) critical points under this definition.

So in a calculus setting, yes I would call $x = \pm 1$ critical points, just for ease. But the bigger lesson is perhaps that there is no absolute truth to terms like "critical points," and this is why it is so important for mathematicians to unambiguously define the terminology they work with.

Addendum: If you have a discontinous function on some interval $[a,b]$ and wish to find extrema, it then becomes very important that you include candidate $x$ values which are not in the domain of $f'(x)$. Think about a cusp maximum, for example. In that example, not being in the domain of $f'$ is not a problem at all.

Integral fan
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    I agree with the definition but how does it imply that every endpoint is a critical point? The endpoints of $\sin x$ on $[0,2\pi]$ have a nonzero derivative. – ryang Jun 18 '22 at 06:48
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    Good point, but I was thinking more on these lines. $\sin(x)$ on $[0,2\pi]$ is the restriction of the sine function onto a closed interval. As long as you define the function $f$ as a map from $[a,b] \to \mathbb{R}$, the derivative will not exist at the endpoints; it no longer inherits the derivative of the full function defined on the reals. This is probably not satisfactory, though? Maybe in a calculus course. – Integral fan Jun 18 '22 at 19:23
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There doesn't seem to be a consensus about whether a boundary point on a domain can be a critical point of a function. See the answer to Boundary point & critical point of a function and Andrew's answer to Is it correct to say all extrema happen at critical points but not all critical points are extrema?.

subrosar
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