Mathematics Asked by In-finite on December 10, 2020

Let $a,$ $b,$ $c,$ $d,$ $e,$ $f$ be nonnegative real numbers.

(a) Prove that

$$(a^2 + b^2)^2 (c^4 + d^4)(e^4 + f^4) ge (ace + bdf)^4.$$

(b) Prove that

$$(a^2 + b^2)(c^2 + d^2)(e^2 + f^2) ge (ace + bdf)^2.$$

I’m not sure how I should start approaching both problems. I believe I should use Cauchy-Schwartz, but I’m not sure.

Any help would be appreciated! Thanks in advance.

These are both CS inequality applications, you should try yourself. Here is the first one:

- $(c^4+d^4)(e^4+f^4) geqslant (c^2e^2+d^2f^2)^2$
- $(a^2+b^2)(c^2e^2+d^2f^2) geqslant (ace+bdf)^2$

Now combine the two to get what you want.

Correct answer by Macavity on December 10, 2020

For (a),

Just use Holder's Inequality.

For (b),

Expand $(a^2 + b^2) (c^2 + d^2) (e^2 + f^2) $. (It is equivalent to the LHS of my answer)

Expand $(ace + bdf)^2$. (It is equivalent to the RHS of my answer)

The problem is just proving
$$
a^2 c^2 e^2 + a^2 c^2 f^2 + a^2 d^2 e^2 + a^2 d^2 f^2 + b^2 c^2 e^2 + b^2 c^2 f^2 + b^2 d^2 e^2 + b^2 d^2 f^2 ge a^2 c^2 e^2 + b^2 d^2 f^2 + 2abcdef
$$ or
$$
a^2 c^2 f^2 + a^2 d^2 e^2 + a^2 d^2 f^2 + b^2 c^2 e^2 + b^2 c^2 f^2 + b^2 d^2 e^2 ge 2abcdef quad textrm{By cancelling 2 terms on the left and right side.}
$$
Now, that is just
$$
(acf - bde)^2 + a^2 d^2 e^2 + a^2 d^2 f^2 + b^2 c^2 e^2 + b^2 c^2 f^2 ge 0
$$
We have proved by this that the inequality is valid for any real numbers, not only non-negative ones.

Answered by Book Of Flames on December 10, 2020

2 Asked on December 20, 2020 by akash-jain

1 Asked on December 20, 2020 by liz

2 Asked on December 20, 2020

diagonalization eigenvalues eigenvectors linear algebra linear transformations matrices

1 Asked on December 20, 2020 by sunit-das

3 Asked on December 20, 2020 by chunky-norris

2 Asked on December 20, 2020 by user842445

0 Asked on December 20, 2020 by masmath

1 Asked on December 20, 2020 by ispring-quiz

1 Asked on December 19, 2020 by latus_rectum

2 Asked on December 19, 2020 by grompchompz

1 Asked on December 19, 2020 by enock-kabibi

4 Asked on December 19, 2020 by shiran-shaharabani

1 Asked on December 19, 2020 by xuuserac

2 Asked on December 19, 2020 by akash-karnatak

0 Asked on December 19, 2020 by isak

1 Asked on December 19, 2020 by stakmod

compactness convergence divergence metric spaces real analysis sequences and series

2 Asked on December 19, 2020 by perfectoid

2 Asked on December 18, 2020 by sammy

0 Asked on December 18, 2020 by peng-yu

Get help from others!

Recent Questions

Recent Answers

- Peter Machado on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP