Prove the inequality (a^3-c^3)/3≥abc((a-b)/c+(b-c)/a)

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In summary, the purpose of proving this inequality is to show that the expression (a^3-c^3)/3 is always greater than or equal to the expression abc((a-b)/c+(b-c)/a), for any values of a, b, and c that satisfy the conditions of the inequality. To prove this inequality, algebraic manipulation and properties of inequalities can be used. This inequality holds true for any values of a, b, and c as long as a is greater than or equal to b, and b is greater than or equal to c. It can be applied to any type of numbers, including real numbers, integers, and fractions, and has real-world applications in fields such as economics, physics, and engineering
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lfdahl
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Let $a, b$ and $c$ be non-zero real numbers, and let $a\ge b \ge c$. Prove the inequality:

$$\frac{a^3-c^3}{3} \ge abc\left(\frac{a-b}{c}+\frac{b-c}{a}\right)$$

When does equality hold?

Source: Nordic Math. Contest
 
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lfdahl said:
Let $a, b$ and $c$ be non-zero real numbers, and let $a\ge b \ge c$. Prove the inequality:

$$\frac{a^3-c^3}{3} \ge abc\left(\frac{a-b}{c}+\frac{b-c}{a}\right)$$

When does equality hold?

Source: Nordic Math. Contest

because $a >= b$ so $(a-b)>0$ or $(a-b)^3>= 0$ or $a^3-3a^2b+3ab^2-b^3>=0$
or $a^3-b^3 >= 3ab(a-b)$
similarly $b^3-c^3 >=3bc(b-c)$
adding we $a^3 -c^3 >= 3abc(\frac{a-b}{c}+\frac{b-c}{a})$
dividing both sides by 3 we get the result

they are equal if a=b =c

rationale

equal if $(a-b)^3 + (b-c)^3 = 0$ sum of 2 non negative numbers when each is zero
 
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  • #3
So simple... (Bow)

-Dan
 
  • #4
Thankyou, kaliprasad for a clever solution. (Clapping)
 

FAQ: Prove the inequality (a^3-c^3)/3≥abc((a-b)/c+(b-c)/a)

1. What is the purpose of proving this inequality?

The purpose of proving this inequality is to show that it holds true for all values of a, b, and c. This can help in solving mathematical problems and making accurate conclusions in various fields of science.

2. How can this inequality be proven?

This inequality can be proven by using mathematical techniques such as algebraic manipulation and substitution. The steps involved in the proof may vary depending on the approach used, but the end result should be the same.

3. What are the implications of this inequality?

The implications of this inequality are that the left side of the equation is always greater than or equal to the right side. This can be useful in comparing different values and determining their relationships.

4. Can this inequality be applied to real-life situations?

Yes, this inequality can be applied in various real-life situations, such as in economics, physics, and engineering. It can help in making predictions and decisions based on mathematical models and calculations.

5. Are there any exceptions to this inequality?

No, this inequality holds true for all values of a, b, and c. However, it is important to note that the inequality is only valid when all the variables are positive numbers.

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