Although this topic may not contain research level mathematics, perhaps the perspective of a researcher is useful in creating new ways of presenting it. After many years of teaching the MVT is various ways unsuccessfully, I came to consider it more useful and practical to try to convey the corollaries of the MVT rather than the statement. If the statement is taught I made it an assertion in words rather than just a formula. I.e. my students struggled unsuccessfully to recall whether the statement contained $f(b)-f(a)$ or $f(a)-f(b)$, whereas I really wanted them to be able to apply it to solve antiderivative problems. Even those students who got the formula right had trouble deducing its meaning. It was also a challenge for students to use inequalities to find intervals on which the derivative was always positive. Hence I developed a presentation, including all proofs if desired, that do not involve manipulating formulas or inequalities. In particular, one consequence is that even the statement of the MVT is not needed at all until one does antiderivative problems.

I agree that what and how to teach depend on ones goals. The narrow goals I set for my first semester course are for students to learn four basic principles, stated in words, and their simplest applications.

1) (“Intermediate value theorem”) the continuous image of an interval is again an interval, applied to existence of solutions of equations.

2)(“Max min value theorem”) the continuous image of a closed bounded interval is again closed and bounded, with applications to existence of solutions of closed interval max min problems.

3)(“Rolle theorem”) A differentiable function is monotone on an interval not containing a critical point, applied to graphing problems and open interval max min problems.

4) (Cor. of MVT) Two differentiable functions with the same derivative in an interval differ there by a constant, with applications to problems such as finding areas by antiderivatives.

Only the last statement requires the mean value theorem, and that only in the proof. In particular I consider statement 4) more important for students to know than the usual statement of the MVT. The deep results, involving completeness, are the first two. Principle 3), including the easy corollary that a function increases on intervals where its derivative is positive, requires only the Rolle theorem. Here is a possible presentation along those lines.

The first two principles, the intermediate value theorem and the max min theorem are probably best assumed in a non honors course, but can be proved without least upper bounds, by constructing recursively an infinite decimal that satisfies the desired theorem.
Using 1), the third principle is equivalent to Rolle’s theorem. I.e. Rolle says a differentiable function taking the same value twice has derivative zero in between, and by principle 1) any function that changes direction takes the same value twice. Hence a differentiable function is strictly monotone on any interval not containing a critical point, and it follows easily that it increases where $f’>0$. It also follows easily that a function whose derivative changes sign changes direction hence has a critical point, so the derivative cannot change sign without becoming zero, i.e. derivatives have the intermediate value property. Thus checking monotonicity requires only finding, usually finitely many critical points, and evaluating the derivative at one point between successive critical points, not on a whole interval. It is significantly easier for students to apply this finitistic process of graphing. The MVT is not needed for any of this.

If one wants to prove principle 4), the relevant statement equivalent to MVT is that two differentiable functions $f,g$ that agree with each other at two points must also have the same derivative somewhere in between. This of course is immediate from Rolle applied to $f-g$. Then since every function $f$ agrees at any two points $a,b$ with some linear function (the secant), and the only linear function with derivative zero is constant, it follows that a function with derivative zero everywhere agrees at any two points with a constant function, hence is itself constant. This implies 4) as usual by subtraction.

This suggests, to me at least, that the formula in the usual MVT is only a needlessly belabored restatement of the numerical slope of a line, which adds nothing to the usefulness of the theorem. I do not argue that anyone else should agree with these ideas. I mention them as a suggestion for those who wish to minimize the use of MVT for some of its usual applications. I.e. the central and deep result is the max min value theorem and the basic fact that an interior extremum is a critical point, as noted above, and the easy corollary stated in the Rolle theorem suffices for all applications short of antiderivative problems. I admit that in a later course where Taylor series enter, the explicit formula begins to look useful.

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