View source for N - 1 Equal Value Principle

The '''n - 1 Equal Value Principle''', or '''n - 1 EV''' for short, is a useful fact regarding the optimization of sums of values of twice continuously differentiable functions with only one inflection point when the sum of the input values is constant.  
==Statement==
Say we have <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> where <math>f</math> is twice continuously differentiable and only has one inflection point.  When <math>x_1, x_2, x_3, ..., x_n</math> are real numbers such that <math>x_1 + x_2 + x_3 + ... + x_n = S</math> for some constant <math>S</math>, <math>f(x_1) + f(x_2) + f(x_3) + ... + f(x_n)</math> is maximized or minimized only if <math>n-1</math> of the <math>x</math>-values are equal.
==Proof==
The following proof is adapted from Mildorf's ''Olympiad Inequalities''. <math>\newline</math>
Give me any sequence of real numbers <math>x</math> and I show that my sequence, with <math>n-1</math> equal terms, gives a greater or equal result.  (Obviously, similar reasoning applies when minimizing the sum.) <math>\newline</math>
WLOG assume that for some value <math>k</math>, <math>f</math>'s inflection point, <math>f</math> is convex for values below and at <math>k</math> and concave otherwise and at <math>k</math>.  Also assume that <math>x</math> are non-increasing.  Now, define <math>m</math> so that <math>x_1 \geq ... \geq x_m \geq k \geq x_{m+1} \geq ... \geq x_n</math>. <math>\newline</math>
Since <math>(x_1, ..., x_m) \prec (x_1 + ... + x_m - (m - 1)k, k, ..., k)</math> and <math>f</math> is convex below and at <math>k</math>, we can apply Karamata's Inequality to obtain: <math>\newline</math> 
<cmath>f(x_1) + ... + f(x_m) \leq f(x_1 + ... + x_m - (m - 1)k) + (m - 1)f(k)</cmath> <math>\newline</math>
Since <math>(k, ..., k, x_{m + 1}, ..., x_n) \succ (\frac{(m-1)k + x_{m + 1} + ... + x_n}{n-1}, ..., \frac{(m-1)k + x_{m + 1} + ... + x_n}{n-1})</math> and f is concave (as opposed to convex) above and at <math>k</math>, we can apply Karamata's Inequality once more to obtain: <math>\newline</math>
<cmath>f(k) + ... + f(k) + f(x_{m + 1}) + ... + f(x_n) \leq (n-1)f(\frac{(m - 1)k + x_{m + 1} + ... + x_n}{n - 1}))</cmath> <math>\newline</math>
Finally, by the transitive property, <math>\newline</math>
<cmath>f(x_1) + ... + f(x_n) \leq f(x_1 + ... + x_m - (m - 1)k) + (n - 1)f(\frac{(m - 1)k + x_{m + 1} + ... + x_n}{n - 1})</cmath> <math>\newline</math>
as desired. <math>\newline</math>
The following video by Evan Chen may be a helpful aid in understanding this technique: https://www.youtube.com/watch?v=Kkhuk8GiuOU