If sum of maximum and minimum value of $y = l o g_{2} (x^{4} + x^{2} + 1) - {log}_{2} (x^{4} + x^{3} + 2 x^{2} + x + 1)$ can be expressed in form $(({log}_{2} m) - n)$ , where $m$ and $n$ are co-prime then compute $(m + n)$ .

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Solution

$y = {log}_{2} {\frac{x^{4} + x^{2} + 1}{x^{4} + x^{3} + 2 x^{2} + x + 1}}$
$= {log}_{2} {\frac{x^{4} + x^{2} + 1}{x^{2} (x^{2} + x + 1) + 1 (x^{2} + x + 1)}}$
$= {log}_{2} {\frac{x^{4} + x^{2} + 1}{(x^{2} + 1) (x^{2} + x + 1)}}$
$= {log}_{2} ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{\frac{(x^{4} + x^{2} + 1)}{x^{2}}}{\frac{(x^{2} + 1)}{x} . \frac{(x^{2} + x + 1)}{x}} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$
$= {log}_{2} ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{x^{2} + 1 + \frac{1}{x^{2}}}{(x + \frac{1}{x}) . (x + 1 + \frac{1}{x})} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$
$= {log}_{2} \frac{{{(x + \frac{1}{x})}^{2} - 1}}{(x + \frac{1}{x}) (x + \frac{1}{x} + 1)}$
$= {log}_{2} ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{(x + \frac{1}{x} - 1) (x + \frac{1}{x} + 1)}{(x + \frac{1}{x}) (x + \frac{1}{x} + 1)} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$
$= {log}_{2} ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{(x + \frac{1}{x} - 1)}{(x + \frac{1}{x})} ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭$
$= {log}_{2} ⎡ ⎢ ⎢ ⎢ ⎣ 1 - \frac{1}{x + \frac{1}{x}} ⎤ ⎥ ⎥ ⎥ ⎦$
$x + \frac{1}{x} ϵ [2, \infty)$
$\Rightarrow \frac{1}{x + \frac{1}{x}} ϵ (0, \frac{1}{2}]$
$\Rightarrow \frac{- 1}{x + \frac{1}{x}} ϵ [\frac{- 1}{2}, 0)$
$\Rightarrow 1 - \frac{1}{x + \frac{1}{x}} ϵ [\frac{1}{2}, 1]$
$\Rightarrow {log}_{2} ⎛ ⎜ ⎜ ⎜ ⎝ 1 - \frac{1}{x + \frac{1}{x}} ⎞ ⎟ ⎟ ⎟ ⎠ ϵ [{log}_{2} \frac{1}{2}, {log}_{2} 1) ϵ [- 1, 0]$
$∴ y_{m a x} = 0$ and $y_{m i n} = - 1$
$∴ y_{m a x} + y_{m i n} = 0 - 1$
$= {log}_{2} 1 - 1$
$\Rightarrow m = 1$ and $n = 1$
$∴ m + n = 2$

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