Least of All
A point moves on a line segment. A function depends on the position
of the point. Where do you expect the point to be for a minimum of
this function to occur.
Age
16 to 18
Challenge level
Problem
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 diagram:
Image
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
Conjectures are 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.
Conjectures are 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.
Getting Started
This could just be a straightforward calculus exercise except that you are asked first to consider the geometric situation and make conjectures about the result before you do any calculus.
To appreciate and fully understand the mathematics you are learning it is good practice to make conjectures if you can and then review your first thoughts in light of further work on the problem.
It is important to ask yourself how many answers you might expect to find (one only or perhaps more than one?) and to ask yourself how you will know if you have found ALL the answers.
Student Solutions
Thank you Ruth from Manchester High School for Girls for your solution to this problem.
Most people's first reaction to this question is exactly the same as Ruth's:
"My initial conjecture was that the minimum value of $f(x)$ is when $x=0$ for any value of $a$, because the function is even and increases as $|x|$ increases."Using calculus, we shall see that this is not so and that the minimum value of $f(x)$ does depend on the value of $a$ .
The given expression is \begin{eqnarray} f(x) &=& (1 + (a+x)^2)(1 + (a-x)^2)\\ &=&(1 + x^2 + a^2 +2ax)(1 + x^2 +a^2 -2ax)\\ &=& (1 + x^2 + a^2)^2 - 4a^2x^2 \\ &=& x^4 + 2x^2(1 + a^2) + (1 +a^2)^2 - 4a^2x^2 \\ &=& x^4 + 2x^2(1 - a^2) + (1+a^2)^2. \end{eqnarray}
As this is a quartic in $x$ there will be one or three turning points.
Differentiating $f$ to find the minima:
$$f'(x) = 4x^3 + 4x(1-a^2) = 4x(x^2 + (1 - a^2))$$
Case 1 : $(1 - a^2) < 0$.
The derivative $f'(x) = 0$ for $x = 0$ and $x = \pm \sqrt (a^2 - 1)$. The second derivative $f''(x) = 12x^2 + 4(1 - a^2) < 0$ at $x = 0$ which gives a maximum value but $f''(x) = 12x^2 + 4(1 - a^2) > 0$ for $x = \pm \sqrt (a^2 - 1)$ giving two minimum points on the quartic where $x = \pm \sqrt (a^2 - 1)$. The minimum value at each point is $f(x) = 4a^2$ where the position of X for these minimum values is clearly dependent of $a$.
Case 2 : $(1 - a^2) > 0$.
The derivative $f'(x) = 0$ if and only if $x = 0$. The second derivative
$f''(x) = 12x^2 + 4(1 - a^2) > 0$ so there is one minimum value $f(x)= (1+a^2)^2$ where the position of X, at $x=0$, is independent of $a$ agreeing with the conjecture.
Case 3 : $(1 - a^2) = 0$.
Note that where $a^2 = 1$ there is a single minimum $f(x) = 4$ at $x=0$ giving continuity between Case 1 and Case 2.
The most likely first conjecture agrees with Case 2 but does not allow the possibility of Case 1. Realising that the function we are minimizing is a quartic we should have taken into account the possibility of two minimum values.
Teachers' Resources
Why do this problem ?
This problem allows students to explore the process of making
mathematical conjectures based on visual intuition. The conjectures
can then be proved/disproved following a straightforward exercise
in finding turning points (whilst taking care over distinguishing
the various cases). This is good training and practice of key
mathematical skills.
Possible approach
Have a class discussion about the positions of X that might
lead to a minimum of the function before anyone actually writes
anything down. Consider and discuss what happens to the function as
X moves along the line segment from P to Q. What conjectures do the
class come up with? Is there a general agreement or are multiple
conjectures stated?
After the students have found the turning points have the
class reflect on whether their findings agree with the original
conjecture(s) and if not why not?
Key questions
- What is a conjecture?
- What features of the geometric situation do you notice?
- How do you expect the function to change as X moves from P to Q?
- Would you expect the position of X giving a minumum value of the function to be dependent or independent of scaling, that is would this position of X be the same relative to the endpoints for a short line segments as when the line segment was stretched to make it longer
and, after doing the calculus:
- What does the graph of the function look like as x varies?
- Does this graph look the same whatever the length of the line segment?
- How many turning points are there?
- Does the nature of the turning points, and/or the position of X where the turning points occur, depend on the value of the constant a?
- Did we miss any of the possibilities when we made our initial conjecture and if so why?
Possible extension
Consider generalisations of this type of problem.
Possible support
Students might struggle with the concept of a conjecture as a
precise mathematical statement which can be proved or disproved
following further investigation. You might like to suggest these
possibilities and to vote on them
- I think that the location of the minimum does not depend on the value of a
- I think that the location of the minimum does depend of the value of a
- I think that the minimum values are always located in the centre of the line.