Find-the-maximum-and-minimum-of-the-expression-i-1-n-a-i-x-i-with-i-1-n-x-i-b-i-2-c-2-where-a-i-b-i-and-c-are-constants-extracted-and-modified-from-Q83331-

Question Number 83341 by mr W last updated on 01/Mar/20

F i n d t h e m a x i m u m a n d m i n i m u m

Double subscripts: use braces to clarify

\underset{i = 1}{\sum^{n}} {(x_{i} - b_{i})}^{2} = c^{2}, w h e r e a_{i}, b_{i} a n d c a r e

c o n s t a n t s .

(e x t r a c t e d a n d m o d i f i e d f r o m Q 83331)

Commented by mr W last updated on 01/Mar/20

S o l u t i o n :

i n n D s p a c e \underset{i = 1}{\sum^{n}} {(x_{i} - b_{i})}^{2} = c^{2} r e p r e s e n t s

a s p h e r e w i t h c e n t e r a t t h e p o i n t

B (b_{1}, b_{2}, \dots, b_{n}) a n d r a d i u s c .

l e t \underset{i = 1}{\sum^{n}} a_{i} x_{i} = s .

\underset{i = 1}{\sum^{n}} a_{i} x_{i} = s r e p r e s e n t s a p l a n e w h i c h

i n t e r c e p t s t h e x_{i} - a x i s a t \frac{a_{i}}{s} .

w h e n t h e p l a n e t a n g e n t s t h e s p h e r e

{w e}^{'} l l g e t t h e m a x i m u m a n d m i n i m u m

v a l u e f r o m s .

i . e . t h e d i s t a n c e f r o m p o i n t B t o t h e

p l a n e s h o u l d b e e q u a l t o t h e r a d i u s o f

t h e s p h e r e .

\frac{∣ \underset{i = 1}{\sum^{n}} a_{i} b_{i} - s ∣}{\sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}}} = c

\Rightarrow \underset{i = 1}{\sum^{n}} a_{i} b_{i} - s = \pm c \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}}

\Rightarrow s = \underset{i = 1}{\sum^{n}} a_{i} b_{i} \pm c \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}}

i . e . \underset{i = 1}{\sum^{n}} a_{i} b_{i} - c \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}} ⩽ \underset{i = 1}{\sum^{n}} a_{i} x_{i} ⩽ \underset{i = 1}{\sum^{n}} a_{i} b_{i} + c \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}}

s p e c i a l c a s e : c = 1 a n d b_{i} = 0

w i t h \underset{i = 1}{\sum^{n}} x_{i}^{2} = 1

- \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}} ⩽ \underset{i = 1}{\sum^{n}} a_{i} x_{i} ⩽ \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}}

Answered by M±th+et£s last updated on 01/Mar/20

Commented by mr W last updated on 01/Mar/20

t h a n k s s i r!

t h i s i s t h e g e n e r a l w a y . b u t i f p o s s i b l e,

i p r e f e r a n o b v i o u s a n d v i s i b l e w a y,

f o r e x a m p l e a g e o m e t r i c a l w a y .

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