Find the Minimum Value of Expression Involving Positive Real Numbers

In summary, the minimum value of the expression (a+b+c+d+e)( \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} +\frac{1}{e}) is 25 if a, b, c, d, and e are positive real numbers. This can be proven using the power mean inequalities, where the arithmetic mean is always greater than or equal to the harmonic mean.
  • #1
matrixone
28
2
if ## a,b,c,d,e ## are positive real numbers, minimum value of [tex](a+b+c+d+e)( \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} +\frac{1}{e} )[/tex]
(A) 25
(B) 5
(C) 125
(D) cannot be determined

My approach :
expanding the expression , i get
[tex]5+a( \frac{1}{b} +\frac{1}{c} +\frac{1}{d} +\frac{1}{e} )+ similar.terms.of.b,c,d,e[/tex]

I can't find any ways to make those expressions vanish ...and some hit and trial gives me ans as 25
but i also can't find any way to make them 20 ... :( ..
or is the answer cannot be determined ?
pls help ...i don't have the answer ...
 
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  • #2
Have you heard of the power mean inequalities? The hint is to make use of the fact that the arithmetic mean is always greater than or equal to the harmonic mean.
(and yes I'm not giving any explicit expressions here because I want you look them up and try to understand them - the Art of Problem Solving wiki is a good place to start)
 
  • #3
thanx a lot for the reference Sir ... So, the direct application of power mean inequality (by putting k1=1 and k2= -1 )gives me the expression is greater than or equal to 25 ... and i hope that is the answer :)
 
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What is the significance of finding the minimum value of an expression involving positive real numbers?

Finding the minimum value of an expression involving positive real numbers is important in many fields of science, such as physics, economics, and engineering. It allows us to determine the smallest possible value that a given system can achieve, which can help us optimize processes and make predictions about the behavior of the system.

How do you find the minimum value of an expression involving positive real numbers?

To find the minimum value of an expression involving positive real numbers, we use mathematical techniques such as differentiation and optimization. By taking the derivative of the expression and setting it equal to zero, we can find the critical points, which are potential minimum values. Then, by plugging these values back into the original expression and comparing them, we can determine the minimum value.

Can the minimum value of an expression involving positive real numbers be negative?

No, the minimum value of an expression involving positive real numbers cannot be negative. This is because the expression only involves positive real numbers, so the smallest possible value it can achieve is zero. If the expression contained negative numbers, the minimum value could potentially be negative.

What are some real-life examples of finding the minimum value of an expression involving positive real numbers?

One example is in economics, where companies use cost functions to determine the minimum cost for producing a certain quantity of goods. Another example is in physics, where scientists use energy functions to determine the minimum amount of energy needed for a system to reach a certain state. In engineering, finding the minimum value of an expression can help optimize designs and processes, such as finding the minimum cost or time for completing a project.

Are there any limitations to finding the minimum value of an expression involving positive real numbers?

Yes, there are some limitations when it comes to finding the minimum value of an expression involving positive real numbers. One limitation is that the expression must be continuous, meaning it has no breaks or gaps, in order for the techniques of differentiation and optimization to be applied. Additionally, the expression must have a finite minimum value, meaning it cannot continue to decrease infinitely.

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