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anemone
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Assume that $x_1,\,x_2,\,\cdots,\,x_n\ge -1$ and $\displaystyle \sum_{i=1}^n x_i^3=0$. Prove that $\displaystyle \sum_{i=1}^n x_i\le \dfrac{n}{3}$.
Is there any reason why we took 4/3 to be multiplied by x^3? I know it simplifies nicely after that. Is the reason because -1 will then be its root?Euge said:Interesting problem. Here is my solution.
It suffices to show that $\max\limits_{x\in [-1,\infty)} (x - \frac43 x^3) = \frac13$, since then we may estimate $$\sum x_i = \sum x_i - \frac{4}{3}\sum x_i^3 = \sum \left(x_i - \frac{4}{3}x_i^3\right) \le \sum \frac{1}{3} = \frac{n}{3}$$ Note that for all $x \ge -1$, \[\begin{align}\frac13 - \left(x - \frac43 x^3\right) &= \frac{4x^3 - 3x + 1}{3}\\ &= \frac{4[(1 + x)^3 - 3(1 + x)^2 + 3(1 + x) - 1] - 3(1 + x) + 4}{3}\\ &= \frac{4(1 + x)^3 - 12(1 + x)^2 + 9(1 + x)}{3}\\ &= \frac{(1 + x)[2(1 + x) - 3]^2}{3}\end{align}\] is nonnegative, and equals zero if $x = -1$ or $x = \frac12$, proving the claim.
DaalChawal said:Is there any reason why we took 4/3 to be multiplied by x^3? I know it simplifies nicely after that. Is the reason because -1 will then be its root?
"Inequality of the sum" refers to the mathematical concept that states the sum of two numbers is always greater than or equal to the difference between the two numbers.
"Inequality of the sum" is used in various fields such as economics, physics, and statistics to compare and analyze data sets, make predictions, and solve equations.
The key principles of "inequality of the sum" include the commutative property, which states that the order of addition does not affect the result, and the transitive property, which states that if a is greater than b and b is greater than c, then a is greater than c.
The difference between "inequality of the sum" and "inequality of the product" is that the former deals with the sum of two numbers while the latter deals with the product of two numbers. Additionally, the inequality of the sum states that the sum is greater than or equal to the difference, while the inequality of the product states that the product is greater than or equal to the difference.
"Inequality of the sum" can be proven using various mathematical techniques such as algebraic manipulation, proof by contradiction, and mathematical induction. It can also be proven using geometric proofs and visual representations.