Proof of $a^{2a}\times b^{2b}\times c^{2c}>a^{b+c}\times b^{c+a}\times c^{a+b}$

In summary, this mathematical proof demonstrates the relationship between exponents and the inequality of a set of numbers, and can be applied in various fields such as economics, physics, and biology. The rearrangement of terms is based on the concept of logarithms and the properties of exponents, and can be generalized for other sets of numbers. However, it has limitations as it only applies to positive real numbers and may not hold true for all sets of numbers.
  • #1
Albert1
1,221
0
given :
$a>b>c>0$
prove :
$a^{2a}\times b^{2b}\times c^{2c}>a^{b+c}\times b^{c+a}\times c^{a+b}$
 
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  • #2
Albert said:
given :
$a>b>c>0$
prove :
$a^{2a}\times b^{2b}\times c^{2c}>a^{b+c}\times b^{c+a}\times c^{a+b}$
hint:
let:
$A=a^{2a}\times b^{2b}\times c^{2c}$
$B=a^{b+c}\times b^{c+a}\times c^{a+b}$
prove:$\dfrac {A}{B}>1$
 
  • #3
Albert said:
hint:
let:
$A=a^{2a}\times b^{2b}\times c^{2c}$
$B=a^{b+c}\times b^{c+a}\times c^{a+b}$
prove:$\dfrac {A}{B}>1$
for $a>b>c>0$ we have :
$\dfrac {A}{B}=a^{(a-b)}a^{(a-c)}b^{(b-a)}b^{(b-c)}c^{(c-a)}c^{(c-b)}$
$=(\dfrac{a}{b})^{a-b}(\dfrac{b}{c})^{b-c}(\dfrac{a}{c})^{a-c}>1^0\times1^0\times 1^0=1$
 
Last edited:

FAQ: Proof of $a^{2a}\times b^{2b}\times c^{2c}>a^{b+c}\times b^{c+a}\times c^{a+b}$

What is the significance of this mathematical proof?

This mathematical proof demonstrates the relationship between exponents and the inequality of a set of numbers. It also shows how these numbers can be rearranged to prove a larger concept.

How can this proof be applied in real-life situations?

This proof can be applied in various fields such as economics, physics, and biology. For example, it can be used to analyze the growth rate of a population or the rate of interest in compound interest calculations.

What is the intuition behind the rearrangement of the terms in this proof?

The rearrangement of terms in this proof is based on the concept of logarithms and the properties of exponents. It shows that when the base of an exponent is greater than 1, the value of the exponent increases as the value of the base increases. This is known as exponential growth.

Can this proof be generalized for other sets of numbers?

Yes, this proof can be generalized for any set of numbers that follow the same pattern of increasing exponents. However, the specific values of the exponents may vary depending on the numbers in the set.

What are the limitations of this proof?

This proof assumes that the numbers in the set are positive real numbers. It also does not take into account any other variables or factors that may affect the outcome of the inequality. Additionally, it only proves the inequality for a specific set of numbers and may not hold true for all sets of numbers.

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