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From: Ray Vickson on 7 Jun 2010 15:54
On Jun 6, 5:22 pm, Albert <albert.xtheunkno...(a)gmail.com> wrote:
> f(x) = (4x^2 - 1)^6, g(x) = (4x^2 - 1)^5
>
> Find {x: f'(x)> g'(x)}
>
> f'(x) = 48x(4x^2 - 1)^5
> g'(x) = 40x(4x^2 - 1)^4
>
> Solve 48x(4x^2 - 1)^5 > 40x(4x^2 - 1)^4
>         48x(4x^2 - 1) > 40x
>          48(4x^2 - 1) > 40
>           6(4x^2 - 1) > 5
>              4x^2 - 1 > 5 / 6
>                  4x^2 > 11 / 6
>                   x^2 > 11 / 24
>         x^2 - 11 / 24 > 0
>   Solve x^2 - 11 / 24 = 0
> x < (-1/2) * sqrt(11/6), x > (1/2) * sqrt(11/6)
>
> This doesn't match with the book's answer - what am I doing wrong?

Simplest approach: look at where L = f'(x) - g'(x) > 0. We have L =
f'(x)-g'(x) = 48*x*(4*x^2-1)^5 - 40*x*(4*x^2-1)^4 =
8*x*(24*x^2-11)*(2*x-1)^4 * (2*x+1)^4. if x is not +-1/2 the last two
factors are > 0 (being even powers of nonzero numbers) so you need
only ask when P = x*(24*x^2-11) > 0. Case 1: both factors in P are >
0: x > 0 and 24*x^2-11 > 0, hence x > sqrt(11/24) =~= 0.667. Case 2:
both factors in P are < 0, hence x < 0 and 24*x^2-11 < 0, hence -
sqrt(11/24) < x < sqrt(11/24). In other words, -sqrt(11/24) < x < 0;
however, we need to exclude the point x = -1/2, so the final result in
Case 2 is -sqrt(11/24) < x < -1/2 union -1/2 < x < 0.

R.G. Vickson
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