The graph of y = ((a + bx)/((x−1)(x−4))) has a turning point at P(2,−1). Find the value of a and b and hence,sketch the curve y = f(x) showing clearly the turning points, asympototes and intercept(s) with the axes.


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Question Number 83964 by Rio Michael last updated on 08/Mar/20

$The graph of$ $y = \frac{a + b x}{(x - 1) (x - 4)}$ $has a turning point at P (2, - 1) . Find the value of a and b$ $and hence, sketch the curve y = f (x) showing clearly the$ $turning points, asympototes and intercept (s) with the$ $axes .$
	Answered by john santu last updated on 08/Mar/20

	$(1) - 1 = \frac{a + 2 b}{1 . (- 2)} \Rightarrow a + 2 b = 2$ $(2) y = \frac{2 - 2 b + b x}{x^{2} - 5 x + 4}$ $y^{'} = \frac{b (x^{2} - 5 x + 4) - (2 x - 5) (2 - 2 b + b x)}{{(x - 1)}^{2} {(x - 4)}^{2}}$ $y^{'} = 0 for x = 2$ $- 2 b - (- 1) (2) = 0$ $2 b = 2 \Rightarrow b = 1 \land a = 0$ $∴ y = \frac{x}{x^{2} - 5 x + 4} .$ $v e r t i c a l a s y m t o t e x = 1 \land x = 4$ $h o r i z o n t a l a s y m t o t e$ $y = \lim_{x \to \infty} \frac{x}{x^{2} - 5 x + 4} = 0$
	Commented by jagoll last updated on 08/Mar/20

	Commented by jagoll last updated on 08/Mar/20

	$this is the graph y = \frac{x}{x^{2} - 5 x + 4}$
	Commented by Rio Michael last updated on 08/Mar/20

	$t h a n k s s i r s g r e a t w o r k$ $i a p p r e c i a t e$
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