Alright, I won't kid around. This problem requires a solution and it has been causing my head some trouble. This is a problem from the Cambridge STEP specimen paper and it goes like this:

Prove that (x^p -1)/p is greater than or equal to (x^q -1)/q.
Where p> q> 0 and 0 < x < 1.

You will have seen the first part of the question which is to explain why ò_a^b f(x) dx > ò_a^b g(x)dx if a < b and f(x) > g(x) for all x in (a,b) - this is clear from areas.

Note that (x^p -1)/p=ò₁^x t^p dt.

For 0 < x < 1, x^p < x^q, so ò_x¹ t^p dt < ò_x¹ t^p dt.

For these integrals the limits are "the right way round" since x < 1.

But ò₁^x t^p dt=-ò_x¹ t^p dt. Thus the required inequality follows.

If this doesn't make sense, sorry - let me know and I'll write it out a bit more clearly.

David

Thanks for the explanation David. However, I guess that I find it difficult to read. Using your method ( or what I can make out of it ), the inequality I end up with gives me an incorrect answer, ( ie:the opposite to what the STEP paper says it should be ). So I would be glad to take you up on your offer of a fuller explanation.

David,

Your method is fine. Just be careful since the question asks for p> q> 0, not p < q, so we obtain the required inequality.

Kerwin