×

# Inequality that is Symmetrical but not Symmetric

1) Show that for any 3 real numbers $$a, b, c$$, we have

$\sqrt{ a^2 - ab + b^2 } + \sqrt{ b^2 - bc + c^2 } \geq \sqrt{ a^2 + ac + c^2 }.$

When does equality occur?

2) If $$a, b$$ and $$c$$ are real numbers such that $$a \geq c, b \geq c, c > 0$$, show that

$\sqrt{c(a-c) } + \sqrt{ c (b-c) } \leq \sqrt{ab} .$

When does equality occur?

3) Let $$a, b$$ and $$c$$ be real numbers with absolute value less than or equal to 1. Show that

$ab+bc+ca + 1 \geq 0.$

When does equality occur?

Hint: I used a tag of #Geometric Interpretation.

Note by Calvin Lin
3 years, 8 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} $$\boxed{123}$$

## Comments

Sort by:

Top Newest

Diagram for (2):

http://i.imgur.com/FXhnSs5.png

- 3 years, 8 months ago

Log in to reply

1) Angle between a and b is 60 deg, angle between b and c is 60 deg and it is a case of triangular inequality.

- 3 years, 8 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...