Unizor - Derivatives - Fermat Theorem

Notes to a video lecture on http://www.unizor.com

Derivatives - Fermat Theorem
(internal local extremums)

First of all, let's talk about terminology.

Internal local extremum is a term used to characterize the behavior of a function defined on some, maybe infinite, interval (a,b) without endpoints (to enable approach to any point of this interval from both sides without restrictions). That's why we use the word internal.

Next word that requires some clarification is local. This word is used to demonstrate that certain characteristics of a function can be observed at some point where it is defined and in the immediate neighborhood of this point. Thus, local maximum (minimum) is a point, where the value of a function is greater (less) than in any other point in some neighborhood of this point, even a very small one.

Finally, extremum is a word that means maximum or minimum. A point where local extremum is attained is called stationary point of a function.

Another important note is that we will consider only differentiable functions, those that have derivatives at each point. Moreover, we assume that these derivatives are continuous functions and, in some cases, differentiable themselves to obtain derivatives of the higher order.
Most functions considered in this course are of this type - polynomial, exponential, logarithmic, trigonometric functions and their combinations.

So, we will talk about local maximum or minimum of sufficiently smooth (in terms of differentiability) functions. This property of smoothness will be assumed by default, even if not explicitly specified.

Fermat Theorem

If a smooth function f(x), defined on interval (a,b), has local extremum at point x0, then its derivative at this point f I(x0) equals to zero:
f I(x0) = 0

Geometrically, since a derivative is related to a tangent of a tangential line to a function, its equality to zero means the horizontal tangential line at a point of local maximum or minimum. The following picture illustrates this.


Proof

Let's consider local maximum first.
Intuitively, local maximum of function f(x) at point x0 means that within some narrow neighborhood of x0, on the left of x0, function f(x) monotonically increases and on the right of it - monotonically decreases.
As was demonstrated before, monotonically increasing functions have non-negative derivative and monotonically decreasing functions have non-positive derivative. That necessitates that at x0 the derivative, a continuous function, as we mentioned above, must be equal to zero.
Here is an illustration:


A little more rigorously, we can assume that a derivative f I(x0) does not equal to zero. So, it's either positive or negative.
As was demonstrated earlier, if a derivative is positive at some point, the function at this point and in the immediate neighborhood of it must be monotonically increasing, thus it cannot have local maximum at this point (values on the left of this point are less then those on the right).
Similarly, if a derivative is negative at some point, the function at this point and in the immediate neighborhood of it must be monotonically decreasing, thus it cannot have local maximum at this point (values on the left of this point are greater then those on the right).
Therefore, a derivative at this point must be equal to zero.

The proof for local minimum is absolutely similar to this.
It would be a nice exercise to right down a proof of it without looking into the proof for maximum offered above.

End of proof.