Question

c/m rằng:

a(a+b)(a+c)(a+b+c)+b²c²\(\ge\)0

Answer 1

Lời giải:
Ta có:

\(A=a(a+b)(a+c)(a+b+c)+b^2c^2=[a(a+b+c)][(a+b)(a+c)]+b^2c^2\)

\(=(a^2+ab+ac)(a^2+ab+ac+bc)+b^2c^2\)

\(=(a^2+ab+ac)^2+bc(a^2+ab+ac)+b^2c^2\)

Đặt \(a^2+ab+ac=x; bc=y\) thì:

\(A=x^2+xy+y^2=(x+\frac{y}{2})^2+\frac{3}{4}y^2\)

Vì \((x+\frac{y}{2})^2\geq 0, y^2\geq 0\Rightarrow A\geq 0\)

Ta có đpcm.