Find range of y=(x/((x−1)(x−2))) .


Question and Answers Forum

All Questions Topic List
Differentiation Questions
Previous in All Question Next in All Question
Previous in Differentiation Next in Differentiation
Question Number 34821 by ajfour last updated on 11/May/18

$F i n d r a n g e o f$ $y = \frac{x}{(x - 1) (x - 2)} .$
Commented by ajfour last updated on 11/May/18

	Answered by ajfour last updated on 11/May/18

	$\frac{d y}{d x} = \frac{x^{2} - 3 x + 2 - x (2 x - 3)}{{(x - 1)}^{2} {(x - 2)}^{2}}$ $\frac{d y}{d x} = 0 \Rightarrow x^{2} = 2 \Rightarrow x = \pm \sqrt{2}$ $y ∣_{x = - \sqrt{2}} = b = \frac{- \sqrt{2}}{(- \sqrt{2} - 1) (- \sqrt{2} - 2)}$ $\Rightarrow b = - \frac{1}{{(\sqrt{2} + 1)}^{2}} = - \frac{1}{(3 + 2 \sqrt{2})}$ $= - (3 - 2 \sqrt{2})$ $y ∣_{x = \sqrt{2}} = a = \frac{\sqrt{2}}{(\sqrt{2} - 1) (\sqrt{2} - 2)}$ $\Rightarrow a = - \frac{1}{{(\sqrt{2} - 1)}^{2}}$ $a = - \frac{1}{3 - 2 \sqrt{2}} = - (3 + 2 \sqrt{2})$ $S o,$ $y \in (- \infty, - 3 - 2 \sqrt{2}] \cup [2 \sqrt{2} - 3, \infty) .$
	Commented by NECx last updated on 12/May/18

	$w o w . . . . T h i s s t y l e i s n e w t o m e .$ $T h a n k s$
	Answered by Joel579 last updated on 11/May/18

	$y (x^{2} - 3 x + 2) = x$ ${y x}^{2} - 3 y x - x + 2 y = 0$ ${y x}^{2} - (3 y + 1) x + 2 y = 0$ $x = \frac{(3 y + 1) \pm \sqrt{{(3 y + 1)}^{2} - 4 y (2 y)}}{2 y}$ $= \frac{(3 y + 1) \pm \sqrt{y^{2} + 6 y + 1}}{2 y}$ $y \neq 0$ $y^{2} + 6 y + 1 ⩾ 0$ $y ⩽ - 3 - 2 \sqrt{2} \lor y ⩾ - 3 + 2 \sqrt{2}$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum