Question

cho a, b, c ∈ [0,1]. CMR: \(a^2+b^2+c^2\le1+a^2b+b^2c+c^2a\)

Answer 1

`a,b,c\in [0;1]`

`=>a(a-1)(b-1)\ge 0`

`<=> a(ab-a-b+1)\ge 0`

`<=> a^2b\ge a^2+ab-a`

Hoàn toàn tương tự:

`=>a^2b+b^2c+c^2a+1\geq a^2+b^2+c^2+ab+bc+ac+1-a-b-c(***)`

Lại có:

`(a-1)(b-1)(c-1)\le 0`

`<=> (ab-a-b+1)(c-1)\le 0`

`<=abc-(ab+bc+ac)+a+b+c-1\le 0`

`<=> ab+bc+ac+1\geq a+b+c+abc\geq a+b+c(******)`

`(***),(******)=> a^2b+b^2c+c^2a+1\geq a^2+b^2+c^2`

Answer 2

bạn tham khảo :https://hoc24.vn/hoi-dap/question/825780.html