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finding the greatest value of a given expression

how to find the greatest value of (a^m)(b^n)(c^p)..... when a+b+c+.... is a constant and m, n, p, ......are positive integers.

Note by Mishti Angel
3 years, 10 months ago

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I have difficulty in understanding what you are saying, and also in interpreting what the rest are saying. My answer would also greatly differ from theirs, and part of that is that your question is not clear. For clarity, can you explain what you are asking for? How many of the \(a, b, c\) terms are you allowed? Just 3, or any finite number? Are the positive integers given? I also believe that you are missing the condition that \(a, b, c \) are non-negative numbers.

For example, if \(a+b+c = k\), then the maximum value of \( a^4 b^2 c^2 \) is actually infinity (without the condition of non-negative numbers). With that condition, a possible value is \( k^4 \) (with \(a=k, b=0, c=0\)), which differs from what Raja and Pratik are saying. (I'm not saying that this is the maximum possible, but you can also track and see where they are going wrong.) Calvin Lin Staff · 3 years, 10 months ago

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THEN YOU HAVE TO KNOW A LITTLE ABOUT THE INEQUALITIES………I THINK I COULD SATISFY YOU…… Let a,b,c…. any positive real numbers whose sum a+b+c+….=given positive integer or a constant…. Since m,n,p,…..are constants (a^m)(b^n)(c^p)…..will be greatest when
(a/m)^m
(b/n)^n(c/p)^p…..[let’s take this product as numbered (1)] is greatest. The product (1) consists of m factors each equal to (a/m),n factors each equal to (b/n),p factors each equal to (c/p),and so on….. the sum of all these factors is equal to m(a/m)+n(b/n)+p(c/p)+…….=a+b+c+…..=constant(given)…. Therefore the product (1) is greatest when all these factors are equal to one another i.e. when a/m=b/n=c/p=…….= [(a+b+c+…..)/(m+n+p+….)] so the greatest value of the product (1) is =(m^m)(n^n)(p^p)…… { [ (a+b+c+…..)/(m+n+p+……)]^(m+n+p+……)} HOPE THIS HELPS….. Raja Metronetizen · 3 years, 10 months ago

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@Raja Metronetizen i came across this method in one of the books....but still wnted to know if there cud be any other way to find out. the max value... Mishti Angel · 3 years, 10 months ago

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@Mishti Angel sorry, any other procedure is not known to me....think any one of this website can help you.... Raja Metronetizen · 3 years, 10 months ago

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@Raja Metronetizen its ok......but if u happen to know or come across any other method.......then kindly let me know .. Mishti Angel · 3 years, 10 months ago

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a,b,c,....are positive integers.... could u pls elaborate ur answer? Mishti Angel · 3 years, 10 months ago

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if a, b, c are positive no's then you can apply AM-GM inequality and note that product of any no of nos is maximum when all the no's are equal. So if, you have to maximize the product, in this case \( a^{m} =b^{n}=c^{p}\) and so on. Pratik Singhal · 3 years, 10 months ago

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Are a,b,c positive integers or they can be any no's?? Pratik Singhal · 3 years, 10 months ago

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