Show there exists constant such that inequality is satisfied for all reals

I don't how to tackle with following problem:

Show there exists constant $0<c<1 $ (depending on $n$) such that $ \sum_{i=1}^n x_i^3x_{i+1} \le c \sum_{i=1}^n x_i^4 $ is satisfied for arbirary reals where $ \sum_{i=1}^n x_i=0 $ and $ x_{n+1}=x_1 $.

I'll be grateful for help

#Algebra

Easy Math Editor

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`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

What have you tried?

Are you familiar with Classical Inequalities? If yes, which do you think would be applicable?

For $n = 2$ , what do you think is the best value of $c$ ?
For $n = 3$ , what do you think is the best value of $c$ ?

Calvin Lin Staff - 5 years, 6 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}`	$\boxed{123}$