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I doubt that the new smash will be more appealing to me than project M, but if it is then I would probably buy a wii u solely for it.

My work ethic was pretty terrible in high school (and even the first year or two of uni tbh).

Why does it suffice to show that cos f(x) doesn't attain a minimum on that small interval? Can't your entire LHS conceivably be negative without this occurring?

And I count using Taylor series as "calculus in disguise" here, because the trig functions are not defined by these series in MX2.

I would be pretty surprised if there was a non-calculus proof of this that wasn't just "calculus in disguise", because you need to look at higher order behaviour of the trig functions (hence the differentiations). Inequalities like sin(x) < x lose too much information.

Does anyone here have a usb controller and like ssb64? (Or melee?) Have been having a blast playing friends online through emulators recently.

...and that constant is zero.

Re: HSC 2014 4U Marathon - Advanced Level There are so many lol. My favourite is f(n)=0.

Unless you are literally allowed to just use the Extreme Value theorem without proof which makes it trivial. (I initially assumed that the thread title referred to the fact that you are asked to prove something quite similar to the EVT.)

Well I am just curious, as I cannot imagine any proof that doesn't involve rigorous definition and study of the reals, which is generally beyond first year.

Hard to say how they want you to prove it without knowing exactly what you are supposed to know about real number / continuous functions already. Do you know Bolzano-Weierstrass or Heine-Borel? Topological characterisations of continuity? Two typical proofs: 1. Continuous functions preserve...

Re: MX2 Integration Marathon .

I certainly didn't.

Re: HSC 2014 4U Marathon - Advanced Level (1+\textrm{cis}(\pi/4))^{2n}=((\textrm{cis}(\pi/8)+\textrm{cis}(-\pi/8))\textrm{cis}(\pi/8))^{2n}\\=(4\cos^2(\pi/8))^n\textrm{cis}(n\pi/4)=(2+\sqrt{2})^n\textrm{cis}(n\pi/4).\\ \\ $Equate real and imaginary parts to finish.$

Re: HSC 2014 4U Marathon - Advanced Level Thanks. Yep, that's right...the last term should be -3xPQR.

Re: HSC 2014 4U Marathon - Advanced Level S=P(x^3)+xQ(x^3)+x^2R(x^3)=c\prod_j (x-\alpha_j).\\ \\ $If $T(x^3):=c^3\prod_j (x^3-\alpha_j^3)=c^3\prod_j (x-\alpha_j)(\omega x-\alpha_j)(\omega^2 x-\alpha_j) \\ \\ = S(x)S(\omega x)S(\omega^2...

Re: HSC 2014 4U Marathon - Advanced Level I think you mean (1/2 + odd) rather than even.

Re: HSC 2014 4U Marathon - Advanced Level Have different remainder upon division by 3. (So when you add P(x^3), xQ(x^3) and x^2R(x^3), the leading terms can't cancel each other out. Otherwise we could not conclude that the degree of the sum is the max of the individual degrees.) Will write...

Re: HSC 2014 4U Marathon - Advanced Level Mine involved the cube roots of unity but is roughly the same length. Note that it is important that the degrees of P(x^3),xQ(x^3) and x^2R(x^3) are distinct mod 3 in order to conclude deg(T)=n, otherwise good solution.

I'm pretty sure that this isn't the universal chord theorem. The universal chord theorem is that the numbers of the form 1/n are the ONLY real numbers such that the above theorem holds. This is a little more subtle and difficult than the question that was asked.