Find-the-local-minimum-value-of-f-x-where-f-x-x-1-x-2-x-1-x-

Question Number 57012 by rahul 19 last updated on 28/Mar/19

F i n d t h e l o c a l m i n i m u m v a l u e o f f (x)

w h e r e f (x) = (x - \frac{1}{x}) + (\frac{2}{x - \frac{1}{x}}) ?

Answered by mr W last updated on 29/Mar/19

l e t t = x - \frac{1}{x}

i f t > 0 :

f (x) = t + \frac{2}{t} ⩾ 2 \sqrt{t \times \frac{2}{t}} = 2 \sqrt{2} = l o c a l m i n

i f t < 0 :

f (x) = t + \frac{2}{t} = - (- t + \frac{2}{- t}) ⩽ - 2 \sqrt{(- t) \times \frac{2}{(- t)}} = - 2 \sqrt{2} = l o c a l m a x

Commented by rahul 19 last updated on 30/Mar/19

thank you sir!

Commented by rahul 19 last updated on 30/Mar/19

S i r, h o w t o d o t h e s a m e p r o b l e m i f

f (x) = (x + \frac{1}{x}) + (\frac{2}{x + \frac{1}{x}}) .

Commented by mr W last updated on 31/Mar/19

l e t t = x + \frac{1}{x}

f o r x > 0 : t ⩾ 2

f o r x < 0 : t ⩽ - 2

f (t) = t + \frac{2}{t}

f^{'} (t) = 1 - \frac{2}{t^{2}} > 0 \Rightarrow f (t) i s s t r i c t l y i n c r e a s i n g

i f t ⩾ 2, i . e . x > 0 :

s i n c s f (t) i s s t r i c t l y i n c r e a s i n g

\Rightarrow m i n . = f (t = 2) = 3

i f t ⩽ - 2, i . e . x < 0 :

s i n c s f (t) i s s t r i c t l y i n c r e a s i n g

\Rightarrow m a x . = f (t = - 2) = - 3

Commented by rahul 19 last updated on 31/Mar/19

S i r, i n t h i s c a s e w e c a n n o t a p p l y

A . M ⩾ G . M b e c a u s e t ⩾ 2 f o r x > 0

u n l i k e t h e a b o v e Q . w h e r e l o c a l m i n i m u m

o c c u r e d a t t = \sqrt{2} ?

Commented by mr W last updated on 31/Mar/19

{t h a t}^{'} s t h e r e a s o n,

y o u a r e a b s o l u t e l y c o r r e c t!

Answered by MJS last updated on 28/Mar/19

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

f^{'} (x) = \frac{x^{6} - 3 x^{4} - 3 x^{2} + 1}{x^{2} {(x^{2} - 1)}^{2}}

x^{6} - 3 x^{4} - 3 x^{2} + 1 = 0

x = \sqrt{y}

y^{3} - 3 y^{2} - 3 y + 1 = 0

y = z + 1

z^{3} - 6 z - 4 = 0

trying factors of - 4 we find

z_{1} = - 2 \Rightarrow y_{1} = - 1 \Rightarrow x_{1} = \pm i

(z + 2) (z^{2} - 2 z - 2) = 0

z_{2} = 1 - \sqrt{3} \Rightarrow y_{2} = 2 - \sqrt{3} \Rightarrow x_{2} = \pm (\frac{\sqrt{6}}{2} - \frac{\sqrt{2}}{2})

z_{3} = 1 + \sqrt{3} \Rightarrow y_{3} = 2 + \sqrt{3} \Rightarrow x_{3} = \pm (\frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2})

f^{″} (x) = \frac{2 (x^{6} + 9 x^{4} - 3 x^{2} + 1)}{x^{3} {(x^{2} - 1)}^{3}}

f^{″} (- \frac{\sqrt{6}}{2} - \frac{\sqrt{2}}{2}) < 0

f^{″} (- \frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2}) > 0 \Rightarrow \min; f (x) = 2 \sqrt{2}

f^{″} (\frac{\sqrt{6}}{2} - \frac{\sqrt{2}}{2}) < 0

f^{″} (\frac{\sqrt{6}}{2} + \frac{\sqrt{2}}{2}) > 0 \Rightarrow \min \Rightarrow f (x) = 2 \sqrt{2}

Commented by rahul 19 last updated on 30/Mar/19

thank you sir!