if-g-x-x-2-x-2x-1-D-g-1-lim-x-g-1-x-ax-b-b-a-a-lt-0-then-find-the-value-of-Max-b-D-g-Domain-

Question Number 146063 by mnjuly1970 last updated on 10/Jul/21

i f g (x) = \frac{x^{2} - x}{2 x - 1}, D_{g} = [1, \infty)

, {l i m}_{x \to \infty} \frac{g^{- 1} (x)}{a x + b} = b - a (a < 0)

t h e n f i n d t h e v a l u e o f M a x (b)

D_{g} = D o m a i n

Answered by gsk2684 last updated on 10/Jul/21

g^{- 1} (x) = y \Rightarrow x = g (y) = \frac{y^{2} - y}{2 y - 1}

\Rightarrow y = \frac{2 x + 1 + \sqrt{4 x^{2} + 1}}{2}

t h e n \lim_{x \to \infty} \frac{2 x + 1 + \sqrt{4 x^{2} + 1}}{2 (a x + b)} = b - a

\lim_{x \to \infty} \frac{2 x + 1 + x \sqrt{4 + \frac{1}{x^{2}}}}{2 (a x + b)} = b - a

\lim_{\frac{1}{x} \to 0} \frac{2 + \frac{1}{x} + \sqrt{4 + \frac{1}{x^{2}}}}{2 (a + \frac{b}{x})} = b - a

\frac{2 + \sqrt{4}}{2 a} = b - a

b = a + \frac{2}{a} ⩽ - 2 \sqrt{a . \frac{2}{a}} (∵ a < 0)

b ⩽ - 2 \sqrt{2}

m a x o f b i s - 2 \sqrt{2}

Commented by mnjuly1970 last updated on 10/Jul/21

v e r y n i c e m r g . s . k t h a n k y o u . .

Commented by gsk2684 last updated on 10/Jul/21

w e l c o m e

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