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Let $a<b<c$ be three real numbers, and $f(x)=(x-a)(x-b)(x-c)$. We want to show that $\int_a^c f(x)\ dx=0$ if and only if $b=\frac{a+c}{2}$.

The first move is to horizontally shift $f$ by $b$ units, that is introduce $g(x)=f(x+b)=(x-r_1)x(x+r_2)$ with $r_1<0<r_2$. Then the statement becomes $\int_{r_1}^{r_2} g(x)\ dx=0$ if and only if $r_1=-r_2$.

If $r_1=-r_2$, the result is true, since the integrand is odd. For the converse, a not difficult but boring computing shows that the integral is $\frac{(r_1-r_2)^3(r_1+r_2)}{12}$, which means that $r_1=-r_2$ (since $r_1-r_2<0$).

This works, but computations does not satisfy me and I have the strong intuition that there could be a more geometric way.

My question: is there a computationless way to show that result?

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    $\begingroup$ Well there is a way to intuitively "see" it, at least in one direction. If b is the midpoint of a and c, then the graph of f reflects itself out of the point (b,0), so it makes sense that the area from a to b is the same as the area from b to c, but negated. You can kind of apply the same logic is the opposite direction by noticing that if b is not the midpoint of a and c then one of the "mounds" formed by the intervals of [a,b] and [b,c] has to be bigger than the other mound, so their areas cannot be the same. $\endgroup$ – ASKASK Oct 28 '15 at 6:14
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    $\begingroup$ Notice that the integral is a strictly monotone (increasing) function of $b$ on $[a,c]$ with opposite signs at $a$ and $c$ - so it has a single root on $[a,c]$ which you already know. $\endgroup$ – A.S. Oct 28 '15 at 6:15
  • $\begingroup$ @A.S.: I intuitively see why the integral is increasing but I fail in proving without explicitly computing it. If you have a way to do it, that would be a nice answer. $\endgroup$ – Taladris Oct 28 '15 at 6:30
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    $\begingroup$ @Taladris Just differentiate the integral with respect to $b$ to see that derivative is positive everywhere - in fact it is constant. $\endgroup$ – A.S. Oct 28 '15 at 6:38
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Try to rewrite it in this way:

$\int_a^c f(x)=\int_a^c (x-a)(x-m)(x-c)dx + (m-b)\int_a^c (x-a)(x-c)dx$

The function $(x-a)(x-m)(x-c)$ is odd function if we place the grid origin in $(m, 0)$. So the integral is zero. The second integral is zero if and only if $m = b$.

Does this proof satisfy you?

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  • $\begingroup$ I like it a lot! It shows the importance of the symmetry, together with using only properties of integrals and quadratic functions. $\endgroup$ – Taladris Oct 28 '15 at 6:32
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    $\begingroup$ Simpson's Rule :For a polynomial $p$ with $deg (p)\leq 3$ we have $\int_a^c p(x) dx=(p(a)+4p((a+c)/2)+p(c))(c-a)/6.$ $\endgroup$ – DanielWainfleet Oct 28 '15 at 6:42
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Note that $f(x) = (x-a)(x-c) < 0$ for all $x \in (a,c)$.

It is not too hard to see (by 'symmetry', since $f(x) = - f({a+c \over 2} -(x-{a+c \over 2})$) that $\int_a^c f(x) (x-{a+c \over 2}) dx = 0$.

Note that $\int_a^c f(x) (x-b) dx = \int_a^c f(x) ((x-b) - (x-{a+c \over 2})) dx = \int_a^c f(x) (({a+c \over 2} -b)) dx$, from which it follows that the integral is zero iff $b={a+c \over 2}$.

Aside: To see why $\int_a^c f(x) (x-{a+c \over 2}) dx = 0$, choose a change of variables $t=a+c-x$. Then we note that $f(t) = f(x)$ and $\int_a^c f(x) (x-{a+c \over 2}) dx =\int_c^a f(t) ({a+c \over 2}-t) (-1) dt = - \int_a^c f(t) (t-{a+c \over 2}) dt$.

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  • $\begingroup$ Very nice solution! $\endgroup$ – ZFR Oct 28 '15 at 6:43
  • $\begingroup$ @RFZ: Thanks! ${}{}$ $\endgroup$ – copper.hat Oct 28 '15 at 6:44
  • $\begingroup$ Can you explain please how you got that $\int \limits_{a}^{c}f(x)(x-\frac{a+c}{2})dx=0$ $\endgroup$ – ZFR Oct 28 '15 at 6:55
  • $\begingroup$ @RFZ: I added an aside above. It is a little easier to see 'geometrically' by looking at symmetry around ${a+c \over 2}$, but the proof above has the advantage of being quick (and I need to sleep now :-)). $\endgroup$ – copper.hat Oct 28 '15 at 7:15
  • $\begingroup$ Good night! :) Thanks a lot for aside! $\endgroup$ – ZFR Oct 28 '15 at 7:21
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Did you notice, every cubic curve is anti-symmetric with respect to its points of inflection, which can be calculated in your case by double differentiation:

$$ x_I =\frac{a+b+c}{3},y_I=f(\frac{a+b+c}{3})? $$

Shifting the origin of coordinate system to $ x_I,y_I $ brings it into a form

$ y_1 = A x_1( x_1^2 - B^2) $ where $ B = (a_1+c_1)/2 $

which is an odd function, the integral or area under cubic curve vanishes between the new $ x_1=a_1, x_2=c_1. $

If not sufficiently clear, shall explain again.

EDIT2:

Sorry about delay. Recasting the cubic using $ h,k $ displacement symbols.

When 3 roots are real,

The cubic equation of third degree polynomial is taken wlog for discussion of roots as:

$$ y = - (x-a) ( x-b) (x-c) \tag{1} $$ has an inflection point at

$$ x = h = (a+b+c)/3 ; \, \, y = k = - (a + b- 2 c) ( b + c -2 a) ( c + a - 2 b)/27 \tag{2} $$

with a spread $\sigma$ on either side at inflection point level $ y= k: $

$$ \sigma = \pm \sqrt{ (a^2 + b^2 + c^2 - a b - b c - c a) /3}\tag{3} $$

i.e., roots when taken as

$$ x = ( h -\sigma, h, h + \sigma ),\,\, y = k \tag{4} $$

bring the cubic equation to another algebraic form equivalent to (1):

$$ ( y-k) = - ( x -h -\sigma)( x- h)( x- h + \sigma) \tag{5} $$

When the roots are in arithmetic progression, letting $ ( a + c) = 2\, b, X = x -h \tag{6} $

it assumes a much simpler form:

$$ y = - X ( X^2 - \sigma ^2) \tag{7} $$

which is an odd function, anti-symmetric with respect to displaced coordinates $X=0 $ .

The integral vanishes when evaluated between $ X = \pm \sigma $ limits. It is same as saying that between three equi-spaced points $ (a, (a+c)/2, c) $ the integral should also vanish in the un-shifted situation of wavy cubic.

The equivalence of cubic forms (1), (5) is valid when there are one real and two complex conjugate roots.

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  • $\begingroup$ This is the most geometrical solution though hardly comprehensible beyond the first paragraph. $\endgroup$ – A.S. Oct 28 '15 at 7:04
  • $\begingroup$ Thanks. Shall modify it. Left out details to the OP as an exercise... $\endgroup$ – Narasimham Oct 28 '15 at 7:10
  • $\begingroup$ I don't like just answars, I like notes and ideas for the answer , +1 $\endgroup$ – Nizar Oct 28 '15 at 8:35
  • $\begingroup$ Nice answer. What does "anti-symmetric" means? Just invariant by rotation of angle $\pi$? $\endgroup$ – Taladris Oct 31 '15 at 4:56
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    $\begingroup$ Meaning an anti-symmetric function about $x= \sigma$ after shift. $\endgroup$ – Narasimham Nov 1 '15 at 4:12
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To expand on Narasimham's answer we need to also show that

$$x_I = (a+b)/2$$if$$c = (a+b)/2$$

Substituting and simplifying gives:

$$x_I = \frac{a+b + \frac{a+b}{2}}{3} = \frac{2(a+b) + (a+b)}{6} = \frac{a+b}{2}$$

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