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How Many Solutions?

Find all the solutions to the this equation.

Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Sine Problem

In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.


Age 16 to 18
Challenge Level

For this question you may like to use a computer graph drawing application or a graphic calculator. For $t = -1/2$, $1/2$ and $2$, sketch graphs of $y = [1 + (x - t)^2][1 + (x + t)^2]$.

You can download the shareware program Graphmatica for free from here as NRICH is an approved distributor of this program. You can find more information about the program from

Then try other values of the parameter $t$. You will see that these graphs have 'different shapes'. Suppose the parameter $t$ varies, then the general shape of the graph varies continuously with $t$.

You can get this far without calculus but you'll probably need calculus to find the turning points and to show that the graph always has a shape similar to the examples above and to find the values of $t$ at which there are transitions from one shape to another.