Editing 2042: Rolle's Theorem

{{comic
| number    = 2042
| date      = September 5, 2018
| title     = Rolle's Theorem
| image     = rolles_theorem.png
| titletext = I mean, if it's that easy to get a theorem named for you ... "a straight line that passes through the center of a coplanar circle always divides the circle into two equal halves." Can I have that one? Wait, can I auction off the naming rights? It can be the Red Bull Theorem or the Quicken Loans Theorem, depending who wants it more.
}}

==Explanation==
{{incomplete|Go a little bit more into the explanation.Explain the museum reference. Do NOT delete this tag too soon.}}

In mathematics, a {{w|differentiable function}} is a function that is "smooth" everywhere, without any sudden breaks or pointy "kinks" or similar. The derivative of such a function is a new function that represents the "slope" or "rate of change" of the original. The function in this comic curves up from point (a) until a point above (c), smoothly turns around, and then curves down from (c) to (b). As a result, the derivative of this function is positive from (a) to (c), and then is negative from (c) to (b). At (c) itself, the function is "flat": the more one zooms in, the more horizontal it looks. The function is moving neither up nor down, so the derivative is neither positive nor negative, but zero. This is what ''f'(c) = 0'' means, as ''f''' is a common notation for the derivative of the function ''f'' in {{w|differential calculus}}.

A {{w|theorem}} in mathematics is a statement that has been ''proven'' from former accepted statements, like other theorems or {{w|axiom}}s. This comic references {{w|Rolle's theorem}}. The theorem essentially states that, if a smoothly changing function has the same output at two different inputs, then it must have one or more turning points in between, as the derivative is zero at each one. As a special case, should the function remain flat between the two inputs, then its derivative is actually zero for every point between the inputs. To [[Randall]], this is obvious. However, the proof of this theorem is not as obvious as the result.

The seeming triviality of the theorem, coupled with the honour bestowed on the theorem namer, leads Randall to make a comparison to attendees of art museums who look at abstract art pieces and perceive only an apparent technical simplicity in the work. Such a visitor might exclaim "My child could paint that!". However, such works of art typically are seen as having value from attributes other than the painterly difficulty in achieving the piece. For example, an artist's work in this style may be lauded for its visionary qualities, or the emotions expressed through the choice of colours or textures. One such artist is {{w|Jackson Pollock}}. The 'clueless' visitor does not see these aspects and believes their child could imitate the piece. Randall suggests he experiences a similar feeling looking at Rolle's Theorem and noting only the obvious correctness without acknowledging the complicated nature of the proof, or other hidden aspects of the theorem.  

In the title text, Randall mentions a line together with a ''coplanar'' circle. This simply means that both those two-dimensional objects must lay in the same plane in a higher, three-or-more-dimensional space. And by this means, every line drawn through the center of a circle is just a diameter which divides it into two equal parts. Even if this fact is trivial, {{w|Proclus}} says that the first man who proved it was {{w|Thales of Miletus|Thales}}. Auctioning of {{w|naming rights}}, also noted in the title text, refers to the practice of naming entertainment venues for companies which pay for the privilege, such as any of the three {{w|Red Bull Arena}}s or {{w|Quicken Loans Arena}}. (See [https://www.reddit.com/r/math/comments/pgj3og/are_there_any_theoremsobjects_involving_company/ "Are there any theorems/objects involving company names."] on r/math.) Furthermore, "Rolle's" sounds like "Rolls", a common abbreviation for the {{w|Rolls-Royce Motor Cars|Rolls Royce}} brand implying possible sponsorship by the British car manufacturer. The naming of mathematical theorems (along with lemmas, equations, laws, methods, etc.) is [http://www.maa.org/external_archive/devlin/devlin_09_05.html not always straightforward] and {{w|List of misnamed theorems|often results in misleading names}}.

Randall implies that there are many seemingly easy theorems like this. For instance the Dirichlet's box principle, also known as the {{w|Pigeonhole principle}}, that states that if you have more objects than containers, you're going to have to put at least two objects in one container.

==Transcript==
:[A single framed picture shows a colored x-y-graph with a text above:]
:'''Rolle's Theorem'''
:<small>From Wikipedia, the free encyclopedia</small>

:Rolle's theorem states that any real, differentiable function that has the same value at two different points must have at least one "stationary point" between them where the slope is zero.

:[The graph shows a sine like curve in blue intersecting the x-axis at points "a" and "b" marked in red while in the middle a point "c" has a vertical dashed green line to the apex and on top also in green f'(c)=0 is drawn with a horizontal tangent line.]

:[Caption below the frame:]
:Every now and then, I feel like the math equivalent of the clueless art museum visitor squinting at a painting and saying "c'mon, my kid could make that." 

{{comic discussion}}

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