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inequalities

Note by Deep Chanda
4 years, 1 month ago

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It is minimum when a1=a2=a3=a4=a5=a6=a7=6/7 Which gives (6/7)^7 / { 1-6/7}^7 = (6/7)^7 / (1/7)^7 =6^7 So, ab=6*7=42 Saurav Shakya · 4 years, 1 month ago

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@Saurav Shakya GM-HM inequality states that for positive reals \(a_i\) then \(\sqrt[n]{a_1a_2 \ldots a_n}\geq \frac{n}{\frac {1}{a_1}+\frac{1}{a_2} \ldots \frac{1}{a_n}}\) where equality occurs when all \(a_i\) are equal. So the value is always greater or equal than \(\frac{7}{\frac {1-a_1}{a_1}+\frac{1-a_2}{a_2} \ldots \frac{1-a_n}{a_n}}\) to the power of seven and its minimum is when this equality occurs. Here I assume them as positive reals less than 1, because if not, then let some \(a_k>1\), then the value is negative and grows smaller to negative infinity as the choice of \(a_k\) grow bigger. Yong See Foo · 4 years, 1 month ago

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@Yong See Foo nicely done Yong See F. Deep Chanda · 4 years, 1 month ago

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@Saurav Shakya I assumed 0 < ai < 1 because if any one ai=0 and any one ai>=1 then we can get - infinity Saurav Shakya · 4 years, 1 month ago

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