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JaneFairfax wrote:

1. andesite

2. subgroups

3. benthic

1. pumice

2. normal

3. abyss

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 - granite

2 - Laplace

3 - submarines

1 - rocks

2 - measure

3 - acoustics

TheDude wrote:

We can also see that 3 < x = a/b < 4.

How?

We can prove that

is increasing by contradiction.Suppse it is not. Then there exist

inNow

and .Also

is either or .Thus we have either

or . This contradicts the fact that bothTherefore

is increasing onYou can prove that

is also increasing on the intervalOops, sorry.

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.

Correct answer below. No peeking until you've made some effort yourself on your homework.

Yoda rocks!

JaneFairfax, how did you get the answer to (xxvii)?

sumpm1 wrote:

1. A function :R->R has a PROPER RELATIVE MAXIMUM VALUE at c if there exists d>0 such that f(x)<f(c) for all x that satisfy 0<|x-c|<d. Prove that the set of points at which has a proper relative maximum value is countable.

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 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. )

1(b)

Hence the LHS is minimized

Use the same trick for 1(c), rewriting

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.

JaneFairfax wrote:

JaneFairfax wrote:

Q: While the musician was at the supermarket, what were her children doing?

A: Playing Haydn seek.

I thought they were playing fluteball.

for all integers *n*. The principal value is when *n* = 0.

mathsyperson wrote:

I *think* this result can be generalised so that instead of f' and f'', you have f[sup](m)[/sup] and f[sup](n)[/sup], where m and n are respectively odd and even.

You'd also need f to be differentiable max(m,n) times rather than just twice.Haven't worked out a solid proof yet though.

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.