Inequality of Four Variables: Prove Σab(a^2+b^2+c^2)≤2

In summary, the conversation discusses the problem of proving that the expression $ab(a^2+ b^2 + c^2) + bc(b^2+ c^2+ d^2) + cd(c^2+ d^2+ a^2) + da(d^2+ a^2+ b^2)$ is less than or equal to $2$, given that the variables $a,\,b,\,c,$ and $d$ are non-negative real numbers and their sum is equal to $2$. The solution to this problem has been requested but not yet provided.
  • #1
anemone
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Let $a,\,b,\,c$ and $d$ be non-negative real numbers such that $a + b + c + d = 2$.

Prove that $ab(a^2+ b^2 + c^2) + bc(b^2+ c^2+ d^2) + cd(c^2+ d^2+ a^2) + da(d^2+ a^2+ b^2) ≤ 2$.
 
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  • #2
anemone said:
Let $a,\,b,\,c$ and $d$ be non-negative real numbers such that $a + b + c + d = 2$.

Prove that $ab(a^2+ b^2 + c^2) + bc(b^2+ c^2+ d^2) + cd(c^2+ d^2+ a^2) + da(d^2+ a^2+ b^2) ≤ 2$.

Please post the solution you have ready. (Time) (Wasntme)

You'd think I would've let you slide on this given that you've posted so many problems, but that's just not how I roll. (Bandit)
 

FAQ: Inequality of Four Variables: Prove Σab(a^2+b^2+c^2)≤2

1. What is the significance of proving Σab(a^2+b^2+c^2)≤2 in terms of inequality of four variables?

The inequality Σab(a^2+b^2+c^2)≤2 is a fundamental result in the study of inequality of four variables. It provides a mathematical framework for understanding the relationships between four variables and their products. It also has implications in various fields such as economics, physics, and engineering.

2. How is the inequality Σab(a^2+b^2+c^2)≤2 related to other inequalities in mathematics?

The inequality Σab(a^2+b^2+c^2)≤2 is a special case of the Cauchy-Schwarz inequality, which states that the square of the sum of products of two sets of numbers is less than or equal to the sum of the squares of the products of the individual numbers in each set. It is also related to the AM-GM inequality, which states that the arithmetic mean of a set of non-negative numbers is greater than or equal to their geometric mean.

3. What are the applications of Σab(a^2+b^2+c^2)≤2 in real-world problems?

The inequality Σab(a^2+b^2+c^2)≤2 has applications in various fields such as optimization, economics, and physics. For example, it can be used to optimize the production of goods given limited resources, to determine the optimal distribution of wealth in a society, and to analyze the stability of physical systems.

4. How is the inequality Σab(a^2+b^2+c^2)≤2 proved?

The inequality Σab(a^2+b^2+c^2)≤2 can be proved using various methods such as algebraic manipulation, induction, and Cauchy-Schwarz inequality. The proof involves showing that the sum of the products of the four variables and their squares is less than or equal to the square of their sum, which is equal to 2 in this case.

5. Are there any variations of the inequality Σab(a^2+b^2+c^2)≤2?

Yes, there are variations of the inequality Σab(a^2+b^2+c^2)≤2 that involve different numbers of variables and different powers. For example, there is a three-variable version of the inequality that involves the sum of the products of the three variables and their cubes. There are also versions that involve higher powers and more variables. These variations have their own proofs and applications in mathematics.

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