Can You Prove This Inequality Involving Real Numbers?

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In summary, the conversation discusses proving the inequality $\dfrac {c}{a+b} +\dfrac {a}{b+c} +\dfrac {b}{c+a}\geq \dfrac {3}{2}$ for positive real numbers a,b,c. One approach is to use calculus to find the points of minimum, while another approach is to use a logical argument to show that the minimum occurs when a=b=c and is equal to 3/2. The conversation also mentions that this is known as Nesbitt's inequality and provides a link to different proofs of it.
  • #1
Albert1
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$ a,b,c \in\mathbb{R}^+ , \,\,\text{Prove:}$

$\dfrac {c}{a+b} +\dfrac {a}{b+c} +\dfrac {b}{c+a}\geq \dfrac {3}{2}$
 
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  • #2
Albert said:
$ a,b,c \in\mathbb{R}^+ , \,\,\text{Prove:}$

$\dfrac {c}{a+b} +\dfrac {a}{b+c} +\dfrac {b}{c+a}\geq \dfrac {3}{2}$

Setting $\displaystyle f(a,b,c) = \frac{c}{a+b} + \frac{a}{b+c} + \frac{b}{c+a}$ find the points of minimum imposing $\displaystyle \frac{\partial f}{\partial a} = \frac{\partial f}{\partial b} = \frac{\partial f}{\partial c} = 0$ and then compute f(*,*,*) in these points... Kind regards $\chi$ $\sigma$
 
  • #3
If without the use of calculus,how can we proceed ?
 
  • #4
Albert said:
If without the use of calculus,how can we proceed ?

As 'pure logical' approach to the problem You can consider that the role of the variable a, b and c is exactly the same, i.e. You can swap them and nothing changes. That means that the minimum of f(*,*,*) is when a=b=c and in that case is f(*,*,*)= 3/2...

Kind regards

$\chi$ $\sigma$
 
  • #5
your analysis is reasonable (Yes)
 
  • #6

FAQ: Can You Prove This Inequality Involving Real Numbers?

What is inequality?

Inequality refers to the unequal distribution of resources, opportunities, and privileges among individuals or groups in a society. It can manifest in various forms, such as economic, social, and political inequality.

What causes inequality?

There are many factors that can lead to inequality, including historical and cultural factors, government policies, and individual characteristics such as race, gender, and socioeconomic status. Globalization and technological advancements have also been identified as contributing factors to inequality.

What are the effects of inequality?

Inequality can have negative impacts on both individuals and society as a whole. It can lead to social and economic instability, hinder economic growth, and result in health disparities and unequal access to education and resources. Inequality also perpetuates social divisions and can lead to social and political tensions.

How can inequality be measured?

Inequality can be measured using various metrics, such as the Gini coefficient, which measures income distribution within a population, or the Human Development Index, which takes into account factors such as education, health, and income. Other measures include the poverty rate, wealth gap, and social mobility.

What can be done to address inequality?

Addressing inequality requires a comprehensive approach that involves government policies, social programs, and individual actions. This may include measures such as progressive taxation, minimum wage laws, investment in education and healthcare, and promoting diversity and inclusion in the workplace. It also requires addressing systemic issues such as discrimination and promoting equal opportunities for all individuals.

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