f-1-3-R-f-x-1-x-A-1-1-B-1-1-3-B-b-1-b-b-1-Find-i-equation-of-line-AB-ii-equation-of-tangent-T-to-C-f-at-point-with-x-1-b-2-iii-Study-relative-positions-of-L-

Question Number 190260 by alcohol last updated on 30/Mar/23

f : [1, 3] \to R, f (x) = \frac{1}{x}

A (1, 1)

B (1, \frac{1}{3})

B^{'} (b, \frac{1}{b}), b ⩾ 1

F i n d

i . e q u a t i o n o f l i n e {A B}^{'}

i i . e q u a t i o n o f t a n g e n t T^{'} t o C_{f} a t p o i n t

w i t h x = \frac{1 + b}{2}

i i i . S t u d y r e l a t i v e p o s i t i o n s o f L_{A B^{'}}, T^{'} t o C_{f}

Answered by a.lgnaoui last updated on 31/Mar/23

i . e q u a t i o n o f l i n e {A B}^{^{'}}

{A B}^{^{'}} : p o r t i o n o f d r o i t e {A B}^{^{'}} (y_{1} = a^{^{'}} x + b^{^{'}})

v e r i f i e {\begin{cases} a^{^{'}} + b^{^{'}} = 1 \\ {b a}^{^{'}} + b^{'} = \frac{1}{b} \end{cases}

y_{1} = \frac{- x}{b} + \frac{1}{b} + 1

i i) e q u a t i o n o f t e n g e n t e T^{:} t o C_{f} a t

p o i n t : x_{0} = \frac{1 + b}{2}

\frac{{d f}_{(x)}}{d x} = \frac{- 1}{x^{2}} x_{0} = \frac{1 + b}{2} \Rightarrow f^{^{'}} {(x)}^{'} = \frac{- 4}{{(1 + b)}^{2}}

\frac{f (x) - f (x_{0})}{x - x_{0}} = \frac{- 4}{{(1 + b)}^{2}}

y = \frac{4 (\frac{1 + b}{2} - x)}{{(1 + b)}^{2}} + \frac{2}{1 + b} = \frac{2 (1 + b - 2 x + 1 + b)}{{(1 + b)}^{2}}

y_{2} = \frac{- 4}{{(1 + b)}^{2}} x + \frac{4}{1 + b}

i i i) Γ^{^{'}} i s b e t w e n t e n g e n t e {A B}^{'} a n d C_{f}

(\frac{1}{b} > \frac{1}{{(1 + b)}^{2}}) w i t h b > 1

y_{1} - y_{2} = (\frac{- x}{b} + \frac{b + 1}{b}) + (\frac{4}{{(1 + b)}^{2}} x - \frac{4}{1 + b})

= (\frac{4}{{(1 + b)}^{2}} - \frac{1}{b}) x + \frac{{(b + 1)}^{2} - 4 b}{b (1 + b)}

= \frac{- {(b - 1)}^{2}}{b {(1 + b)}^{2}} x + \frac{{(b - 1)}^{2}}{b (1 + b)}

\frac{{(b - 1)}^{2}}{b (1 + b)} [\frac{- x}{b + 1} + 1]

f o r m e a^{^{″}} x + b^{^{″}} (b^{^{″}} > 0 a^{^{″}} < 0)

Prime causes double exponent: use braces to clarify

Commented by a.lgnaoui last updated on 31/Mar/23

Commented by alcohol last updated on 31/Mar/23

t h a n k y o u v e r y m u c h

Facebook Tweet Pin