Find the maximum and minimum of the expression 𝚺_(i=1) ^n a_i x_i with 𝚺_(i=1) ^n (x_i −b_i )^2 =c^2 , where a_i , b_i and c are constants. (extracted and modified from Q83331)


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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

$S o l u t i o n :$ $i n n D s p a c e \underset{i = 1}{\sum^{n}} {(x_{i} - b_{i})}^{2} = c^{2} r e p r e s e n t s$ $a s p h e r e w i t h c e n t e r a t t h e p o i n t$ $B (b_{1}, b_{2}, . . ., b_{n}) a n d r a d i u s c .$ $l e t \underset{i = 1}{\sum^{n}} a_{i} x_{i} = s .$ $\underset{i = 1}{\sum^{n}} a_{i} x_{i} = s r e p r e s e n t s a p l a n e w h i c h$ $i n t e r c e p t s t h e x_{i} - a x i s a t \frac{a_{i}}{s} .$ $w h e n t h e p l a n e t a n g e n t s t h e s p h e r e$ ${w e}^{'} l l g e t t h e m a x i m u m a n d m i n i m u m$ $v a l u e f r o m s .$ $i . e . t h e d i s t a n c e f r o m p o i n t B t o t h e$ $p l a n e s h o u l d b e e q u a l t o t h e r a d i u s o f$ $t h e s p h e r e .$ $\frac{∣ \underset{i = 1}{\sum^{n}} a_{i} b_{i} - s ∣}{\sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}}} = c$ $\Rightarrow \underset{i = 1}{\sum^{n}} a_{i} b_{i} - s = \pm c \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}}$ $\Rightarrow s = \underset{i = 1}{\sum^{n}} a_{i} b_{i} \pm c \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}}$ $i . e . \underset{i = 1}{\sum^{n}} a_{i} b_{i} - c \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}} ⩽ \underset{i = 1}{\sum^{n}} a_{i} x_{i} ⩽ \underset{i = 1}{\sum^{n}} a_{i} b_{i} + c \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}}$ $s p e c i a l c a s e : c = 1 a n d b_{i} = 0$ $w i t h \underset{i = 1}{\sum^{n}} x_{i}^{2} = 1$ $- \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}} ⩽ \underset{i = 1}{\sum^{n}} a_{i} x_{i} ⩽ \sqrt{\underset{i = 1}{\sum^{n}} a_{i}^{2}}$
	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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