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In James Stewart's Calculus (8th edition) Fermat's Theorem is stated as

If $f$ has a local maximum or minimum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$.

The author cautions that the converse of the statement is generally false. In other words, finding $x$-values such that $f'(x) = 0$ does not necessarily imply that a local max or min occurs there. The functions $f(x) = x^3$ and $f(x) = |x|$ are provided as examples.

The author then introduces the idea of critical numbers and defines them as

A critical number of a function $f$ is a number $c$ in the domain of $f$ such that either $f'(c) = 0$ or $f'(c)$ does not exist [$f'(c)$ is undefined]

and then modifies Fermat's Theorem as

If $f$ has a local maximum or minimum at $c$, then $c$ is a critical number of $f$.

My confusion begins when the author describes how to find absolute max and min.

To find an absolute maximum or minimum of a continuous function on a closed interval, we note that it is either local (in which case it occurs at a critical number by [Fermat's modified Theorem]) or it occurs at an endpoint of the interval[...]

The wording to me seems like the author is suggesting to use critical points to find or identify local maxes and mins of a function. This notion seems to be further reinforced in the first step of the Closed Interval Method

  1. Find the values of $f$ at the critical numbers of $f$ in $(a, b)$

I don't think this is what the author meant since he goes to great length to caution against the converse of Fermat's Theorem, but I'm not entirely sure if rephrasing Fermat's Theorem in terms of critical numbers now renders the converse true.

Just to be safe I will ask the following questions:

  1. Is the author suggesting that we use critical points to find local maxes and mins? If so, wouldn't that mean the converse of Fermat's modified Theorem is true?

  2. Is the fact that local maxes and mins occur at critical numbers a consequence of Fermat's modified Theorem or its converse?

I suspect the answer is no to the first question, and the fact that local maxes and mins occur at critical points is a consequence of Fermat's modified Theorem which implies that if a local max or min exists, then it will always occur at a critical value. Therefore, finding critical values is only finding potential spots where local maxes or mins might occur.

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The author states that if $f'(c)$ exists, then it must be $0$. Notice that in your examples, it did not exist.

To answer your question:

Is the author suggesting that we use critical points to find local maxes and mins?

He is.

If so, wouldn't that mean the converse of Fermat's modified Theorem is true?

No. As you mentioned earlier, there are critical points at which the maximum or minimum may not occur at.

Is the fact that local maxes and mins occur at critical numbers a consequence of Fermat's modified Theorem or its converse?

I will explain this to you in a different way. First, you must convince yourself that if $f$ is continuous on an interval $[a,b]$, then $f$ must attain a minimum and a maximum on $[a,b].$ There are three possible scenarios:

  • The extrema value occurs at some point $c \in (a,b)$ such that $f'(c) = 0$;

  • The extrema value occurs at some point $c \in (a,b)$ such that $f'(c)$ does not exist;

  • The extrema value occurs at $f(a)$ or $f(b)$. In other words, the endpoint(s).

We call all possible points $c$ in all three of the cases above critical points of $f$. Now, this is only tells us where the critical points are. This implies that if we want to find the maximum or minimum value of $f$ on some interval, then we still have to check all of those points.

Therefore, finding critical values is only finding potential spots where local maxes or mins might occur.

This is correct.

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