Question

Find the critical points of the following functions and test them for their maxima and minima.
$f\left(x\right)=\frac{x^2-3x+2}{x^2+3x+2}$

Answer 1

$f\left(x\right)=\frac{x^2-3x+2}{x^2+3x+2}=\frac{(x-2)(x-1)}{(x+2)(x+1)}{f}^{\prime }\left(x\right)=\frac{(2x-3)(x^2+3x+2)-{(x}^2-3x+2\left)\right(2x+3)}{({x^2+3x+2)}^2}{f}^{\prime }\left(x\right)=0$

$2x^3+6x^2+4x-3x-9x-6=2x^3-3x^2-5x+6$

$6x^2=12$

$x=\pm \sqrt{2}$

$f\left(\sqrt{2}\right)=\frac{(\sqrt{2}-2)(\sqrt{2}-1)}{(\sqrt{2}+2)(\sqrt{2}+1)}=-17+12\sqrt{2}$

$f(-\sqrt{2})=\frac{(-\sqrt{2}-2)(-\sqrt{2}-1)}{(-\sqrt{2}+2)(-\sqrt{2}+1)}=-17-12\sqrt{2}$