Question

cho a,b,c ∈ [0 ; 1]. Cmr: \(a+b^2+c^3-ab-bc-ca\le1\)

Answer 1

\(0\le a;b;c\le1\Rightarrow a+b^2+c^3\le a+b+c\)

\(\Rightarrow a+b^2+c^2-ab-ac-bc\le a+b+c-ab-bc-ca\) (1)

Mặt khác cũng do \(0\le a;b;c\le1\)

\(\Rightarrow\left(1-a\right)\left(1-b\right)\left(1-c\right)\ge0\)

\(\Leftrightarrow1-abc-a-b-c+ab+bc+ca\ge0\)

\(\Leftrightarrow a+b+c-ab-bc-ca\le1-abc\le1\) (2)

(1);(2) \(\Rightarrow a+b^2+c^3-ab-bc-ca\le1\)

Dấu "=" xảy ra khi \(\left(a;b;c\right)=\left(1;0;0\right)\) và hoán vị