Question

Tìm GTNN của A = x^3 + y^3 + xy biết x+ y =1 B= (x-1)^2 +(x-3)^2

Answer 1

\(A=\left(x^3+y^3+xy\left(x+y\right)\right)-xy\left(x+y\right)+xy\)

=>    \(A=\left(x+y\right)\left(x^2+y^2\right)-xy.1+xy\)

=>   \(A=x^2+y^2-xy+xy\)

=>    \(A=x^2+y^2\ge\frac{\left(x+y\right)^2}{2}=\frac{1^2}{2}=\frac{1}{2}\)

DẤU "=" XẢY RA <=>    \(x=y\). MÀ    \(x+y=1\)

=> A min \(=\frac{1}{2}\Leftrightarrow x=y=\frac{1}{2}\).

Answer 2

\(B=x^2-2x+1+x^2-6x+9\)

=>   \(B=2x^2-8x+10\)

=>   \(B=2\left(x^2-4x+4\right)+2\)

=>   \(B=2\left(x-2\right)^2+2\)

CÓ:    \(2\left(x-2\right)^2\ge0\forall x\Rightarrow2\left(x-2\right)^2+2\ge2\)

=>   \(B\ge2\)

DẤU "=" XẢY RA <=>    \(2\left(x-2\right)^2=0\Leftrightarrow x=2\)

VẬY B MIN = 2 <=>    \(x=2\)