# Describing Mean Value Theorem as "Fundamental Theorem Of Differential Calculus"

Referring to "Introduction to Real Analysis" 4th Ed (2011) by Bartle and Sherbert. (ch 6, sec 2, page 174)

In fact the Mean Value Theorem is a wolf in sheep's clothing and is the Fundamental Theorem of Differential Calculus.

What could have lead them to state this? All the other calculus books (Stewart, Apostol, Thomas...) have given special prominence to this theorem. I am definitely not confusing this with Fundamental Theorem of Calculus. What I mean is : What could have lead to this heightened significance of this theorem?

Although this could be a subjective opinion, I would love to receive some expansive insights.

I further request relative comparison of all the significant theorems of differential calculus namely, Extreme value, Mean value and Intermediate value theorem.

PS: I emphasize that this question is completely disassociated with FTOC. If someone feels like the intent of author in the quote above is ambiguous/subjective, they may feel free to disregard it in their answers.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Mathematics Meta, or in Mathematics Chat. Comments continuing discussion may be removed. Jan 8 at 12:46

This question might get closed due to "Opinion-Based" thoughts on it.

The "Best" we can come up with is the "Opinion" of the Original Authors.

Consider that text book itself to get some ideas.

Page 172 : "The Mean Value Theorem, which relates the values of a function to values of its derivative , is one of the most useful results in real analysis"

Page 173 has the Short Statement of the theorem , while the Proof is hardly 4-5 Sentences long.
Proof involves the geometric view & utilizes this Image :

Page 174 then has the remark :
"The geometric view of the Mean Value Theorem is that there is some point on the curve $$y = f (x)$$ at which the tangent line is parallel to the line segment through the points $$(a, f (a))$$ and $$(b, f (b))$$ . Thus it is easy to remember the statement of the Mean Value Theorem by drawing appropriate diagrams. While this should not be discouraged, it tends to suggest that its importance is geometrical in nature, which is quite misleading. In fact the Mean Value Theorem is a wolf in sheep’s clothing and is the Fundamental Theorem of Differential Calculus. In the remainder of this section, we will present some of the consequences of this result. Other applications will be given later."
"The Mean Value Theorem permits one to draw conclusions about the nature of a function $$f$$ from information about its derivative $$f'$$ . The following results are obtained in this manner."

## What the Authors are trying to convey :

(1) Even though the Proof utilized geometry here , MVT is important & useful everywhere , not just in geometry.
(2) The applications & consequences of MVT are numerous & very useful in a wide variety of Situations.

The Authors then list out 4 theorems/lemmas/Extrema-tests , all with short Statements & Proofs using MVT.

High-lighting the importance of MVT , the Authors talk about (A1) Alternating roots of $$\sin$$ & $$\cos$$ (A2) Similarity with the roots of Bessel functions , (B) Using MVT to calculate square roots like $$\sqrt{105}$$ , then conclude with Common Inequalities including (C1) Bernoulli , (C2) Exponential function , (C3) trigonometric functions . . . .

There are more applications further in the textbook.
Praising & elevating MVT is then very natural to the Authors here.

When the Authors move on to the "Fundamental Theorem of Calculus" [ Page 216 ] , the Proof involves MVT [ Page 217 ] in the Core Idea , the rest are just elementary calculations . . . .

In that view , "MVT" might be termed "obviously" Equivalent to "FTOC" , hence "MVT" should be given due importance too , like "FTODC" . . . .

OP : What could have lead to this heightened significance of this theorem ?
My thoughts : Elementary Nature of the Statement , Simplicity of the Proof , Pervasiveness of the Applications , Direct Utility in the FTOC Proof are some of the Keys , nudging the Authors to make that Claim.

Prem's answer is much more comprehensive than mine, but one reason to view the Mean Value Theorem (MVT) as a "fundamental" result in differential calculus is because it is the first theorem one proves which, for a function $$f\colon [a,b]\to\mathbb R$$ relates the "infinitesimal" information about $$f$$ given by the values $$f'(x)$$ $$(x\in [a,b])$$ to, if you like "macroscopic" information about $$f$$: a simple example being that if $$f'(x)> 0$$ for all $$x\in [a,b]$$ the $$f$$ is increasing.

Although familiarity with the statement makes this result seem easy, if you try to prove this starting directly from the definition of the derivative, you quickly discover that if $$c \in [a,b]$$ then $$f'(c)>0$$ only implies that if $$x$$ is sufficiently close to $$c$$, then $$(x-c)$$ and $$f(x)-f(c)$$ have the same sign, i.e. $$x and $$c for all $$x$$ with $$|x-c|$$ sufficiently small. Indeed there exist differentiable functions $$f\colon \mathbb R\to \mathbb R$$ for which $$f'(0)=1$$ but $$f$$ is not increasing on any interval containing $$0$$.