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I have the following three functions

$f_1(x) = \frac{1}{4} (8-3x + \sqrt{(x-2) (5x-14)}) (1-x)$

$f_2(x) = \frac{1}{8} (12-4x + \sqrt{2} \sqrt{(5x-14)(x-3)} + \sqrt{2} \sqrt{(x-2)(x-3)} )(1-x)$

$f_3(x) = (x-1)(x-2)$

How possible can it be shown that

$f_1(x) > (or <) f_2(x) < (or >) f_3(x)$

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For specific $x$? Or as $x\to+\infty$? –  Hagen von Eitzen Oct 3 '12 at 13:13
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Hard to parse. You have written $f_1$ in the form $1/4ab$. Is that $(1/4)ab$? or $1/(4ab)$? or $(1/(4a))b$? –  Gerry Myerson Oct 3 '12 at 13:14

1 Answer 1

If you are interested as $x \to \infty$, you can just look for the highest power of $x$. For $f_1(x)$, the square root goes as $\sqrt 5x$, so the whole thing goes as $(3-\sqrt 5)x^2$. You can do similarly with the second and the third is $x^2$. This says that as $x \to \infty, f_3(x) \gt f_1(x)$

If you are interested in all $x$, note that they all are $0$ at $x=1$, so none is strictly greater than any other.

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+1 This method is optimal. –  mick Oct 3 '12 at 15:22

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