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Question Number 92885 by unknown last updated on 09/May/20

Commented by unknown last updated on 09/May/20

$a + b + c = 3 . Find the minimum value of A$
Commented by mr W last updated on 09/May/20

$A_{m i n} = 3 a t a = b = c = 1$
Commented by prakash jain last updated on 10/May/20

$If negative values allowed :$ $a = 102, b = 1, c = - 100 ?$ $I also think question implictely$ $assume + ve values .$
Commented by unknown last updated on 10/May/20

$If using A M - G M sir ? ?$
Commented by mr W last updated on 10/May/20

$a, b, c c a n b e n e g a t i v e . b u t s i n c e b o t h$ $t h e f u n c t i o n a n d t h e c o n s t r a i n t a r e$ $s y m m e t r i c f o r a, b a n d c, t h e e x t r e m$ $v a l u e o f t h e f u n c t i o n m u s t l i e o n$ $t h e s y m m e t r y a x i s, i . e . a = b = c .$
Commented by prakash jain last updated on 10/May/20

$a = 102, b = 1 c = - 100$ $a + b + c = 102 + 1 - 100 = 3$ $\frac{{(2 - 102)}^{3}}{102} + \frac{(2 - 1^{3})}{1} + \frac{(2 - {(- 100)}^{3})}{- 100} =$ $- 20403 < 3$
Commented by prakash jain last updated on 10/May/20

$Hi unknown,$ $AM - GM can only be used if$ $the question mentions condition$ $that a, b, c > 0 .$ $A M - G M$ $\frac{a + b}{2} ⩾ \sqrt{a b}$ $d o e s n o t h o l d f o r$ $a = - 2, b = - 8$
Commented by mr W last updated on 10/May/20

$i t h i n k i f a n e x t r e m v a l u e (m i n i m u n$ $o r m a x i m u m) e x i s t s, i t m u s t l i e o n$ $t h e s y m m e t r y a x i s . i n c u r r e n t$ $c a s e a t a = b = c = 1 t h e f u n c t i o n g e t s$ $a l o c a l e x t r e m (m i n i m u m) v a l u e .$ $a n g l o b a l m a x i m u m o r m i n i m u m$ ${d o e s n}^{'} t e x i s t, s i n c e w e s e e i n$ $A = 2 (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) - (a^{2} + b^{2} + c^{2})$ $A \to + \infty i f a o r b o r c \to 0^{+} a n d$ $A \to - \infty i f a o r b o r c \to 0^{-} .$
Commented by unknown last updated on 11/May/20

$I^{'} m sorry sir,$ $i^{'} m sorry sir, I forgot to write if a, b, c is p o s i t i v e number . Thank you very much sir . I^{'} m sorry$
Commented by prakash jain last updated on 11/May/20

$Ok . Have you been able to prove$ $it usimg AM - GM or you need help ?$
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