# High powered Inequality

Prove $x^4+y^4+z^4\geq xyz(\sqrt{xy}+\sqrt{yz}+\sqrt{xz})$.

Note by Joshua Ong
3 years, 11 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by:

Use AM-GM.

- 3 years, 11 months ago

The result follows immediately from Muirhead: $$(4,0,0)\succ \left(\dfrac{3}{2},\dfrac{3}{2},1\right)$$

- 3 years, 11 months ago

Except it is not given that $$x,y,z>0$$ and Muirhead's inequality holds for positive reals.

- 3 years, 11 months ago

Are $$x,y,z$$ positive reals or just reals? (the same question for some of the other problems. You should provide the given restrictions on $$x,y,z$$ since otherwise we're not quite sure what they are).

- 3 years, 11 months ago