f-x-x-a-x-b-x-a-x-b-Find-minimum-and-maximum-

Question Number 53732 by ajfour last updated on 25/Jan/19

f (x) = \frac{(x + a) (x + b)}{(x - a) (x - b)}

F i n d m i n i m u m a n d m a x i m u m .

Commented by mr W last updated on 25/Jan/19

a t x = a a n d x = b : f (x) \to \pm \infty

Commented by ajfour last updated on 25/Jan/19

T h e r e i s a d e f i n i t e a n s w e r g i v e n,

m a x i m u m a n d m i n i m u m a r e

a s k e d . I t s f r o m a g o o d b o o k, S i r .

m u s t t h e n b e l o c a l m i n i m u m a n d

m a x i m u m .

Commented by mr W last updated on 25/Jan/19

s i n c e f (x) \to \pm \infty w h e n x \to a o r \to b,

t h e r e i s n o m i n . o r m a x . b u t

t h e r e c a n b e a l o c a l m i n i m u m a n d a

l o c a l m a x i m u m .

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

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

f^{'} (x) = 0

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

x^{2} - (a + b) x + a b = x (2 x - (a + b))

x^{2} = a b

\Rightarrow x = \pm \sqrt{a b}

i . e . i f a b ⩾ 0 t h e r e a r e l o c a l m i n . a n d

l o c a l m a x . a t x = \pm \sqrt{a b}

f {(x)}_{m i n / m a x} = \frac{(a \pm \sqrt{a b}) (b \pm \sqrt{a b})}{(a \mp \sqrt{a b}) (b \mp \sqrt{a b})}

i f a b < 0 t h e r e i s n o l o c a l m i n . o r m a x .

b u t w e {d o n}^{'} t n e e d t o u s e c a l c u l u s t o

g e t t h i s r e s u l t, j u s t l o o k a t

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

i f a b ⩾ 0,

x + \frac{a b}{x} ⩾ 2 \sqrt{a b}

x + \frac{a b}{x} ⩽ - 2 \sqrt{a b}

f (x) = 1 + \frac{2 (a + b)}{x + \frac{a b}{x} - (a + b)} ⩽ 1 + \frac{2 (a + b)}{2 \sqrt{a b} - (a + b)} = - {(\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}})}^{2} = f_{m a x}

f (x) = 1 + \frac{2 (a + b)}{x + \frac{a b}{x} - (a + b)} ⩾ 1 + \frac{2 (a + b)}{- 2 \sqrt{a b} - (a + b)} = - {(\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}})}^{2} = f_{m i n}

Commented by ajfour last updated on 25/Jan/19

a n s w e r : - {(\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}})}^{2} & - {(\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}})}^{2} .

Commented by mr W last updated on 25/Jan/19

t h i s i s t h e s a m e a s m y r e s u l t (s e e b e l o w) .

a n s w e r i n b o o k i s n o t t h e b e s t, s i n c e

i t r e q u e s t s t h a t a ⩾ 0 a n d b ⩾ 0, b u t i n

f a c t t h i s i s n o t n e c e s s a r y . i t i s o n l y

n e c e s s a r y t h a t a b ⩾ 0, e . g . a = - 2, b = - 5 .

Commented by mr W last updated on 25/Jan/19

\frac{(a + \sqrt{a b}) (b + \sqrt{a b})}{(a - \sqrt{a b}) (b - \sqrt{a b})}

= \frac{\sqrt{a} (\sqrt{a} + \sqrt{b}) \sqrt{b} (\sqrt{b} + \sqrt{a})}{\sqrt{a} (\sqrt{a} - \sqrt{b}) \sqrt{b} (\sqrt{b} - \sqrt{a})}

= - \frac{(\sqrt{a} + \sqrt{b}) (\sqrt{a} + \sqrt{b})}{(\sqrt{a} - \sqrt{b}) (\sqrt{a} - \sqrt{b})}

= - {(\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}})}^{2} = a s g i v e n i n b o o k

\frac{(a - \sqrt{a b}) (b - \sqrt{a b})}{(a + \sqrt{a b}) (b + \sqrt{a b})}

= \frac{\sqrt{a} (\sqrt{a} - \sqrt{b}) \sqrt{b} (\sqrt{b} - \sqrt{a})}{\sqrt{a} (\sqrt{a} + \sqrt{b}) \sqrt{b} (\sqrt{b} + \sqrt{a})}

= - \frac{(\sqrt{a} - \sqrt{b}) (\sqrt{a} - \sqrt{b})}{(\sqrt{a} + \sqrt{b}) (\sqrt{a} + \sqrt{b})}

= - {(\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}})}^{2} = a s g i v e n i n b o o k

Commented by ajfour last updated on 25/Jan/19

y e s, s i r . t o o w e l l d i d y o u e x p l a i n .

T h a n k y o u t o o m u c h .

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Jan/19

y = \frac{(x + a) (x + b)}{(x - a) (x - b)}

l n y = l n (x + a) + l n (x + b) - l n (x - a) - l n (x - b)

\frac{1}{y} \frac{d y}{d x} = \frac{1}{x + a} + \frac{1}{x + b} - \frac{1}{x - a} - \frac{1}{x - b}

f o r m a x / m i n \frac{d y}{d x} = 0

\frac{1}{x + a} - \frac{1}{x - b} + \frac{1}{x + b} - \frac{1}{x - a} = 0

\frac{x - b - x - a}{(x + a) (x - b)} + \frac{x - a - x - b}{(x + b) (x - a)} = 0

\frac{- (a + b)}{(x + a) (x - b)} + \frac{- (a + b)}{(x + b) (x - a)} = 0

\frac{1}{(x + a) (x - b)} + \frac{1}{(x + b) (x - a)} = 0

x^{2} - a x + b x - a b + x^{2} - b x + a x - a b = 0

2 x^{2} - 2 a b = 0

x = \pm \sqrt{a b}

\frac{1}{y} \frac{d y}{d x} = \frac{1}{x + a} + \frac{1}{x + b} - \frac{1}{x - a} - \frac{1}{x - b}

\frac{d y}{d x} = \frac{(x + a) (x + b)}{(x - a) (x - b)} [\frac{1}{x + a} + \frac{1}{x + b} - \frac{1}{x - a} - \frac{1}{x - b}]

\frac{w a i t \dots}{} w l

n o w i a m g o i n g t o f i n d c h a n g e o f s i g n

u s i n g f i r s t d e r i v a t i v d m e t h o d t o f i n d m a x / m i n

(\frac{d y}{d x}) = \frac{(x + a) (x + b)}{x^{2} - x (a + b) + a b} \times [\frac{1}{x + a} - \frac{1}{x - b} + \frac{1}{x + b} - \frac{1}{x - a}]

= \frac{x^{2} + x (a + b) + a b}{x^{2} - x (a + b) + a b} \times [\frac{x - b - x - a}{x^{2} - x b + a x - a b} + \frac{x - a - x - b}{x^{2} - a x + b x - a b}]

= \frac{N}{D} \times [\frac{- (a + b)}{x^{2} - a b + x (a - b)} + \frac{- (a + b)}{x^{2} - a b - x (a - b)}]

= \frac{N}{D} \times - (a + b) \times [\frac{x^{2} - a b - x (a - b) + x^{2} - a b + x (a - b)}{{(x^{2} - a b)}^{2} - x^{2} {(a - b)}^{2}}]

n o w D < N s o \frac{N}{D} > 1 \frac{N}{D} \times - (a + b) = - v e

[\frac{2 (x^{2} - a b)}{{(x^{2} - a b)}^{2} - x^{2} {{(a + b)}^{2} - 4 a b}]

w h e n x > \sqrt{a b} x^{2} = h + a b

= \frac{N}{D} \times - (a + b) \times [\frac{2 (x^{2} - a b)}{{(x^{2} - a b)}^{2} - x^{2} {{(a + b)}^{2} - 4 a b}]

= (- v e) \times [\frac{2 h}{h^{2} - (h + a b) {(a - b)}^{2}}] = (- v e) \times \frac{+ v e}{- v e}

w h e n x > \sqrt{a b} s i g n c h a n g e f r o m - v e t o + v e

s o a t x = \sqrt{a b} f (x) m i n

y_{} = \frac{(x + a) (x + b)}{(x - a) (x - b)} (a t x = \sqrt{a b})

= \frac{\sqrt{a} (\sqrt{b} + \sqrt{a}) \times \sqrt{b} (\sqrt{a} + \sqrt{b})}{\sqrt{a} (\sqrt{b} - \sqrt{a}) \times \sqrt{b} (\sqrt{a} - \sqrt{b})}

= (-) \frac{{(\sqrt{a} + \sqrt{b})}^{2}}{{(\sqrt{b -} \sqrt{a})}^{2}} \to m i n v a l u e

s o a t x = \sqrt{a b} m i n v a l u e

a t x = - \sqrt{a b} m a x v a l u e

y = \frac{(- \sqrt{a b} + a) (- \sqrt{a b} + b)}{(- \sqrt{a b} - a) (- \sqrt{a b} - b)}

= \frac{\sqrt{a} (\sqrt{a} - \sqrt{b}) \times - \sqrt{b} (\sqrt{a} - \sqrt{b})}{- \sqrt{a} (\sqrt{a} + \sqrt{b}) \times - \sqrt{b} (\sqrt{a} + \sqrt{b})}

= (- 1) \times \frac{{(\sqrt{a} - \sqrt{b})}^{2}}{(\sqrt{a} + \sqrt{b})} \leftarrow m a x v z l u d \dots

Commented by ajfour last updated on 25/Jan/19

w e l c o m e b a c k S i r . .

Commented by ajfour last updated on 25/Jan/19

t h a n k s s i r, d o i p o s t t h e a n s w e r ?

Commented by tanmay.chaudhury50@gmail.com last updated on 25/Jan/19

i w e n t t o s e e t h e n e w s \dots