Let a,b $\in$ $\mathbb{R}.$ Show that $a^4+b^4+8\ge 8ab.$

Question

Let $a,b \in \mathbb{R}.$ Show that $a^4+b^4+8\ge 8ab.$

The question is from the inequalities section of An Excursion in mathematics by Bhaskaraycharya Pratisthanan. My heuristics include using the AM-GM inequality. I am unable to design the problem to proceed further.
Please provide me some hints.

Aditya Hase · Accepted Answer · 2014-11-14T13:26:35.720

14

You're on right path just a small trick using AM-GM Write $$a^4+b^4+8=a^4+b^4+4+4$$ And then apply AM-GM $$\frac{a^4+b^4+4+4}{4}\ge \sqrt[4]{16a^4b^4}$$

$$\frac{a^4+b^4+4+4}{4}\ge 2ab$$

$${a^4+b^4+8}\ge 8ab$$

Or You can just SOS it $$(a^2-b^2)^2\ge0\implies a^4+b^4-2a^2b^2\ge0$$

$$2(ab-2)^2\ge0\implies2a^2b^2+8-8ab\ge0$$ Add above two inequalities to get $$ a^4+b^4+8\ge8ab$$

edited Nov 14 '14 at 13:26

answered Nov 08 '14 at 16:03

Aditya Hase

8,851

You may also add to your answer how the inequality can be obtained from the Rearrangement Inequality. I think it will help the OP more. – Nov 08 '14 at 16:41
@user170039 how can we obtain it from rearrangement inequality ? – r9m Nov 09 '14 at 21:43
1

@r9m: The AM-GM Inequality can be proved from Rearrangement Inequality. – Nov 10 '14 at 03:05
@user170039 thanks for clearing it ! :) – r9m Nov 10 '14 at 04:24

score 4 · Answer 2 · edited Dec 13 '14 at 08:56

4

We have $$\frac{a^4+b^4}{2}\geq a^2b^2$$ by using AM-GM inequality, therefore we will obtain $$a^4+b^4+8-8ab\geq 2a^2b^2+8-8ab=2(ab-2)^2\geq 0$$

edited Dec 13 '14 at 08:56

Anastasiya-Romanova 秀

19,345

answered Nov 08 '14 at 16:13

Dr. Sonnhard Graubner

95,283

6

Do you ever look at your answers and think "that could look a lot better if I had bothered to make it pretty". – Sarah Nov 17 '14 at 21:34
i try to find the answer as fast as i can Sarah – Dr. Sonnhard Graubner Nov 17 '14 at 21:35
7

I think you would be doing this community a great favour by later revising your answer and making it high quality. – Sarah Nov 17 '14 at 21:36
what do you mean with high quality? – Dr. Sonnhard Graubner Nov 17 '14 at 21:37
3

Proper grammar. Good spacing between formulas. Look at other answers as an example. – Sarah Nov 17 '14 at 21:39
thanks Sarah this is an argument, i will think about it – Dr. Sonnhard Graubner Nov 17 '14 at 21:40
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I do not know if you are aware @Dr.SonnhardGraubner but several users have been complaining about the quality of your posts. Posting answers is fine but the ultimate point of SE is to provide high quality questions and answers. Your contribution is welcome as long as you are making an effort to participate. You will find that you will not have as much negative feedback if your answers are well written and formatted as Sarah suggested. – Ali Caglayan Nov 17 '14 at 21:43
ok thanks for your hint – Dr. Sonnhard Graubner Nov 17 '14 at 21:45

score 1 · Answer 3 · answered Dec 19 '14 at 14:57

1

Another way to look at it is to consider the following. $$(a^2-b^2)^2\geq0$$ Therefore

$$ a^4+b^4\geq2a^2b^2$$

Then from there you proceed the same way as Dr.Sonnhard did..

answered Dec 19 '14 at 14:57

math424

56

Let a,b $\in$ $\mathbb{R}.$ Show that $a^4+b^4+8\ge 8ab.$

3 Answers3