### Folium of Descartes

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

### Witch of Agnesi

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

### Quick Route

What is the quickest route across a ploughed field when your speed around the edge is greater?

# Least of All

##### Stage: 5 Challenge Level:

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:
This text is usually replaced by the Flash movie.

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.
Given your insights, can you suggest possible locations for the minimum values of $g(x) = (1+ XP^4)(1+XQ^4)$?