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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

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In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

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Sketch the graphs of y = sin x and y = tan x and some straight lines. Prove some inequalities.


Stage: 5 Challenge Level: Challenge Level:1
This problem also appears on Underground Mathematics, where you can find more materials to support classroom use.

Why do this problem?

This is a quick, simple problem on gradients with a neat result. It will help to reinforce ideas about lines and coordinate geometry and factorising expressions.

Possible approach

This problem is ideally used as a lesson starter. It might be useful when revisiting ideas about coordinate geometry in a more advanced context. The second part might challenge some students and could be left as an optional extra for those who find the first part straightforward.

In the problem, the result is described as 'beautiful'. Do students see it as such? Can they understand why a mathematician might see it as beautiful?

Key question

Can you find the gradients of the segments?

Possible extension

Can you find a similar result if the parabola were replaced by the cubic equation $y=x^3$.

How far can you repeat the analysis if two lines joining two pairs of points on the parabola were perpendicular?

Possible support

Just do the first part of the problem.