Math Is Fun Forum

I get it now. Thanks, TheDude.

Incidentally, are you all (apart from Ricky) only assuming that x is positive? Can't x be negative as well? After all, if you multiply a number four times, you do get a positive number. Ricky's method may allow for negative numbers but I don't really know what you can do with the method.

From all the numbers I've tried, I'm inclined to believe that there is no negative solution. I'm just wondering if you can definitely prove that no negative solutions exist.

1 - basalt

2 - cosets

3 - suboceanic

1 - rocks

2 - measure

3 - acoustics

We can prove that

is increasing by contradiction.

Suppse it is not. Then there exist

in I such that

.

Now

and

.

Also

is either

or

.

Thus we have either

or

. This contradicts the fact that both f and g are increasing on I.

Therefore

is increasing on I.

You can prove that

is also increasing on the interval I by the same method.

Ooh! It looks all so simple once you know the answer that it makes you feel kinda stupid wondering why you never thought of it before.

Thanks.

Lead

Yoda rocks!

This is one of those questions which are intuitively obvious but present such a challenge to prove. I can only suggest using the metric-space properties of the real numbers. Try proving that the set of points at which ƒ has a proper relative maximum value is nowhere dense in R. Countability should follow from the fact that all nowhere-dense subsets of R are countable. (Are they? I think so anyway. )

In theory, the betting game could go on forever, but in practice no game can go on forever. You can't possibly spend an eternity in the casino. Besides, you'll miss your bus.

A more practical phrasing of the question might be, say, what is the probability that you have £50 within 50 bets? Assuming you only have time for 50 bets before your bus arrives.

I thought they were playing fluteball.

Intuitively, f', f'' > 0 means the function is convex and strictly increasing; such a curve should intuitively go to infinity as x goes to infinity. I just wanted a concrete analytical proof of what was intuitive to me.

In fact f' and f'' don't have to be positive on the whole real line, they just have to be positive on the interval [a,∞) for some real number a. This can even be easily shown - just replace 0 by a in the your proof above.