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Analysis help needed

Prove x^4+3x^-4>=2 root 3 for all nonzero x.

Any ideas where to get started here? Is it something to do with a binomial expansion?

Reply 1

Mutleybm1996

Original post by pineapplechemist

Prove x^4+3x^-4>=2 root 3 for all nonzero x.

Any ideas where to get started here? Is it something to do with a binomial expansion?

Proof by induction maybe?

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Reply 2

jameswhughes

Original post by pineapplechemist

Prove x^4+3x^-4>=2 root 3 for all nonzero x.

Any ideas where to get started here? Is it something to do with a binomial expansion?

Try moving the

2\sqrt3

to the other side of the inequality, then see if it looks familiar.

Reply 3

pineapplechemist

Original post by jameswhughes

Try moving the

2\sqrt3

to the other side of the inequality, then see if it looks familiar.

EDIT: got to (x^2-root3x^-2)(x^2+root3x^-2)>=0

(edited 9 years ago)

Edited above post

Look at the equation

f(x) = x^4 + 3x^{-4}

Find any extrema (more specifically, the minima) and then find the values

f

take at these minima.

(edited 9 years ago)

Reply 6

BabyMaths

I'd start from

\left(x^2-\dfrac{\sqrt{3}}{x^2}\right)^2 \ge 0

Reply 7

james22

You can also notice that it is a quadratic in disguise, try letting u=x^4 and note that u must be positive.

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