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

$$f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x$$

You can download the shareware graph drawing program Graphmatica for free from here as NRICH is an approved distributor of this program. You can find more information about the program from http://www.graphmatica.com/