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Also some changes should be made slightly to the terminology, on line 5 it should say "then it takes every value from [0,1] at least twice"

I did not use the hint, I did this instead: \\ $a) Take some$ \ k \ $in$ \ [0,1] \ $since$ \ f \ $is continuous on$ \ [0,1] \ $then is continuous on$ \ [0,k] \ $and$ \ [k, 1] \\ $By the intermediate value theorem then$ \ f \ $takes every value in$ \ [f(0) , f(k)] \ $by the first interval and...

For the second part I am getting a \in \left[0 , \frac{n-1}{n} \right]

Re: HSC 2014 4U Marathon - Advanced Level \\ $The function$ \ g \ $is continuous at$ \ x = a \ $if$ \ \lim_{x \to a} g(x) = g(a) \\ $Taking the first inequality$ \ \ \lim_{x \to a} f(x) \leq \lim_{x \to a} g(x) \leq \lim_{x \to a} h(x) \\ $Since$ \ f \ $and$ \ h \ $are continuous, then$ \\...

Re: HSC 2014 4U Marathon \\ $Find the sum of angles$ \ A, B \ $where$ \ 0 \leq A,B \leq \pi \ \ $and$ \\ \\ \sin A + \sin B = \sqrt{\frac{3}{2}} \\ \cos A + \cos B = \sqrt{\frac{1}{2}}

Re: HSC 2014 4U Marathon - Advanced Level I answered in the thread you posted

\\ $After arriving at the equation$ \ \ \cos 5\theta = 16\cos^5 \theta - 20\cos^3 \theta + 5\cos \theta \\ $To arrive at the polynomial equation in$ \ x \ $it is clear that we must choose$ \ \theta \ $so that$ \ \cos 5\theta = 1 \\ \\ $Therefore the equation$ \ \ 16x^5-20x^3+5x - 1 = 0 \ $has...

Re: HSC 2014 4U Marathon - Advanced Level \\ $Prove via squeeze theorem$ \ \ e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n Someone else post a question please

Re: MX2 Integration Marathon \int_0^{\pi} \frac{\sin^3x}{\sin^2x + 8} \ dx

Re: MX2 Integration Marathon nice work \int_0^{\pi/6} \frac{\sqrt{1 + \sin x}}{\cos x} \ dx

Re: MX2 Integration Marathon \int_0^{\pi/4} \frac{x + \sin x}{1 + \cos x} \ dx

Re: MX2 Integration Marathon Yep well done!

Re: MX2 Integration Marathon Hint: Let u = (that expression)

By tangent-secant theorem AP^2 = AB \cdot AC AQ^2 = AB \cdot AC \therefore \ AP = AQ \\ $Then using the first equation$ \ 12^2 = 6 \cdot AC

Re: MX2 Integration Marathon \int_0^{\pi/2}\frac{x}{\sin x + \cos x} \ dx

Re: HSC 2014 4U Marathon \\ $What is the domain and range for$ \ \ f(x) = \sin^{-1}x + \sin^{-1} \frac{1}{x}

Re: HSC 2014 4U Marathon Handwritten Solution

Re: MX2 Integration Marathon Yep something like that

Re: MX2 Integration Marathon Let's generalize that just a little bit \\ $Find the re-currence formula in$ \ n \ $for$ \\ \\ I_n = \int_0^1 x^n (1-x^m)^{1/k} \ dx \ \ $for$ \ n > (m+1) \ $for naturals$ \ n (its doable I tried it)

Oh, well in that case the diagram shows that P varies on one arm of the hyperbola, meaning that -90 < theta < 90 meaning sec is positive.