A point $X$ moves on the line segment $PQ$ of length $2a$ where
$XP=a+x$, $XQ=a-x$ and $-a\leq x \leq a$, as in the following
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You are interested in finding the minimum value of the function
$f(x)=(1 + XP^2)(1 + XQ^2)$. Without writing anything down can you
suggest where the location of X that gives the minimum value(s) of
$f(x)$ will be? Do you think that this will depend on the value of
$a$? Once you have considered the matter, write your thoughts down
as a clear, precise conjecture.
Check your conjecture using calculus to find the minimum
values. Was this as expected?
Given your insights, can you suggest possible locations for the
minimum values of $g(x) = (1+ XP^4)(1+XQ^4)$?
NOTES AND BACKGROUND
speculations which people try to prove or disprove. Some
mathematical conjectures are so difficult to prove that many
mathematicians, over a hundred years or more, have tried and failed
to prove them. There is prize money and worldwide fame for anyone
who can prove one of these famous conjectures such as The Reimann
Conjecture. This is also called the Reimann Hypothesis which leads
to the question "Is a conjecture the same as a hypothesis?"
The answer is, strictly speaking, no. The word hypothesis is used in pure mathematics
to mean one of the conditions in a theorem, for example 'If a
triangle is right-angled" is a hypothesis in Pythagoras' Theorem. A
very different meaning for the word hypothesis , which is used in
statistics, is a speculation which is reinforced or refuted by the
acquisition of new information. Such hypotheses are not proved
logically true or false but merely considered, in the light of
evidence, to be more likely or unlikely to be true.