for-a-b-c-gt-0-and-a-b-c-2-find-min-2ab-2-b-3-c-2bc-2-c-3-a-2ca-2-a-3-b-or-disprove-that-such-a-minimum-doesn-t-exist-

Question Number 159125 by mr W last updated on 13/Nov/21

f o r a, b, c > 0 a n d a + b + c = 2

f i n d m i n (2 {a b}^{2} + b^{3} c, 2 {b c}^{2} + c^{3} a, 2 {c a}^{2} + a^{3} b)

o r d i s p r o v e t h a t s u c h a m i n i m u m

{d o e s n}^{'} t e x i s t .

Commented by mr W last updated on 14/Nov/21

t h a n k s s i r! t h i s i s e x a c t l y t h a t w h a t

i a l s o t h i n k . t h i s q u e s t i o n w a s r i s e d

d u e t o a l o n g d i s c u s s i o n i n

Q 159035 .

Commented by MJS_new last updated on 14/Nov/21

for a, b, c ⩾ 0 \land a + b + c = 2 the minimum

obviously is zero :

i . e . c = 0 \Rightarrow \min (2 {a b}^{2}, 0, a^{3} b) = 0

but for a, b, c > 0 only the limit exists :

\lim_{c \to 0^{+}} (\min (2 {a b}^{2} + b^{3} c, 2 {b c}^{2} + c^{3} a, 2 {c a}^{2} + a^{3} b)) = 0

\Rightarrow we can give no minimum value, only a

greatest lower bound which is 0

Commented by MJS_new last updated on 14/Nov/21

I saw that \dots

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