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Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

How Many Solutions?

Find all the solutions to the this equation.

After Thought

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

Curve Match

Age 16 to 18 Challenge Level:

The following five functions have been plotted accurately on the chart below
$$y=\sqrt{x}\quad\quad y=x^2\quad\quad y=x\quad\quad y=(x^2+x)/2\quad\quad y=2\sqrt{x}-x$$

Which is the $x$-axis and which is the $y$-axis? Which curves correspond to which functions?

Imagine that you wish to plot a route between $(0, 0)$ and $(1, 1)$. Can you find a function which does this without intersecting any of the existing curves, except at the endpoints?

What other curves can you plot which only intersect the existing curves at the end points?