Activity 1-2: Inequalities

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Activity 1-2 Inequalities
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What inequalities do you know?
What do you think the most basic inequality of
all might be?
Maybe the Triangle Inequality.
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Notice that a triangle has another basic
inequality
a lt b lt c ? A lt B lt C
The length of any one side of a triangle is less
than the sum of the other two.
a lt b c, b lt a c, c lt a b.
Travelling from A to B direct is shorter than
travelling from A to B via C we are saying the
shortest distance between any two points is a
straight line.
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Standard inequalities like these are of great use
to the mathematician.
More arise from this question  How do we find
the average of two non-negative numbers a and b?
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How are these ordered? Does the order of size
depend on a and b?
Task try to come up with a proof that AM GM
for all non-negative a and b. When does equality
hold? Now try to show that GM HM for all
non-negative a and b. When does equality hold?
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We can see that equality only holds in each
case when a b.
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We can often come up with a diagram that
demonstrates an inequality.   What inequality
does the following diagram illustrate?
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(No Transcript)
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How about this?
Hint calculate OA, AB, AC.
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So AM ? GM ? HM.
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Can we prove the AM-GM inequality for three
numbers? That is, if a, b, c gt 0, does 3abc a3
b3 c3 hold?
First reflect on this diagram.
So we have that ab bc ac ? a2 b2 c2.
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Now reflect on this diagram...

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With thanks to Claudi Alsina and Roger B.
Nelsen, authors of When Less is More Visualising
Basic Inequalities.
Carom is written by Jonny Griffiths,
hello_at_jonny-griffiths.net