proof-that-i-1-n-a-i-a-i-x-2015-has-exactly-n-real-roots-o-lt-a-1-lt-a-n-

Question Number 55685 by tm888 last updated on 02/Mar/19

p r o o f t h a t

\underset{i = 1}{\sum^{n}} \frac{a_{i}}{a_{i} - x} = 2015 h a s e x a c t l y n r e a l

r o o t s . o < a_{1} \dots . < a_{n}

Answered by mr W last updated on 03/Mar/19

l e t y = f (x) = \underset{i = 1}{\sum^{n}} \frac{a_{i}}{a_{i} - x}

y^{'} = \underset{i = 1}{\sum^{n}} \frac{a_{i}}{{(a_{i} - x)}^{2}} > 0

\Rightarrow t h e f u n c t i o n i s s t r i c k l y i n c r e a s i n g .

\lim_{x \to a_{k}^{(-)}} f (x) = \underset{i = 1}{\sum^{n}} [\lim_{x \to a_{k}^{(-)}} \frac{a_{i}}{a_{i} - x}]

= \underset{i = 1, \neq k}{\sum^{n}} \frac{a_{i}}{a_{i} - a_{k}} + \lim_{x \to a_{k}^{(-)}} \frac{a_{k}}{a_{k} - x} = + \infty

\lim_{x \to a_{k}^{(+)}} f (x) = \underset{i = 1}{\sum^{n}} [\lim_{x \to a_{k}^{(+)}} \frac{a_{i}}{a_{i} - x}]

= \underset{i = 1, \neq k}{\sum^{n}} \frac{a_{i}}{a_{i} - a_{k}} + \lim_{x \to a_{k}^{(+)}} \frac{a_{k}}{a_{k} - x} = - \infty

t h a t m e a n s f o r x \in (a_{k}, a_{k + 1}) w i t h k = 1, 2, \dots n - 1

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

\lim_{x \to a_{k}} f (x) = \lim_{x \to a_{k}^{(+)}} f (x) = - \infty

\lim_{x \to a_{k + 1}} f (x) = \lim_{x \to a_{k + 1}^{(-)}} f (x) = + \infty

i . e . f (x) = c h a s o n e a n d o n l y r o o t i n (a_{k}, a_{k + 1})

w i t h c = a n y r e a l n u m b e r, e . g . 2015 .

\lim_{x \to - \infty} f (x) = \underset{i = 1}{\sum^{n}} [\lim_{x \to - \infty} \frac{a_{i}}{a_{i} - x}] = + 0

t h a t m e a n s f o r x \in (- \infty, a_{1})

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

\lim_{x \to - \infty} f (x) = + 0

\lim_{x \to a_{1}} f (x) = \lim_{x \to a_{1}^{(-)}} f (x) = + \infty

i . e . w i t h c > 0, e . g . c = 2015

f (x) = c h a s o n e a n d o n l y r o o t i n (- \infty, a_{1})

\lim_{x \to + \infty} f (x) = \underset{i = 1}{\sum^{n}} [\lim_{x \to + \infty} \frac{a_{i}}{a_{i} - x}] = - 0

t h a t m e a n s f o r x \in (a_{n}, + \infty)

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

\lim_{x \to a_{n}} f (x) = \lim_{x \to a_{1}^{(+)}} f (x) = - \infty

\lim_{x \to + \infty} f (x) = - 0

i . e . w i t h c > 0, e . g . c = 2015

f (x) = c h a s n o r o o t i n (a_{n}, + \infty)

s u m m a r y :

f (x) = c > 0, e . g . c = 2015 h a s

o n e a n d o n l y o n e r o o t i n (- \infty, a_{1})

o n e a n d o n l y o n e r o o t i n (a_{k}, a_{k + 1}) w i t h k = 1, 2, \dots, n - 1

n o r o o t i n (a_{n}, + \infty)

t o t a l l y f (x) = c h a s e x a c t l y n r o o t s .

Commented by otchereabdullai@gmail.com last updated on 03/Mar/19

The great and ideal professor W

Commented by mr W last updated on 03/Mar/19

Commented by mr W last updated on 03/Mar/19

w e c a n s e e

f (x) = \underset{i = 1}{\sum^{n}} \frac{a_{i}}{a_{i} - x} = 0 h a s e x a c t l y n - 1 r o o t s a n d

f (x) = \underset{i = 1}{\sum^{n}} \frac{a_{i}}{a_{i} - x} = c \neq 0 h a s e x a c t l y n r o o t s .

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