inequalities

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 - 8 years, 4 months ago

I assumed 0 < ai < 1 because if any one ai=0 and any one ai>=1 then we can get - infinity

saurav shakya - 8 years, 4 months ago

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 - 8 years, 4 months ago

nicely done Yong See F.

Deep Chanda - 8 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$