for a, b, c >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.


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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

$I saw that . . .$
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