Question

1. Cho a>0, b>0, x>0> Tìm GTNN của \(A=\frac{\left(x+a\right)\left(x+b\right)}{x}\)

2. Tìm x, y nguyên thoả mãn \(x^2+x-y^2=0\)

Answer 1

\(A=\frac{x^2+\left(a+b\right)x+ab}{x}=x+\frac{ab}{x}+a+b\)

\(\Rightarrow A\ge2\sqrt{\frac{ab.x}{x}}+a+b=2\sqrt{ab}+a+b\)

Dấu "=" xảy ra khi \(x=\sqrt{ab}\)

b/ \(x^2+x=y^2\)

- Với \(x=0\Rightarrow y=0\)

- Với \(x\ge1\Rightarrow\left\{{}\begin{matrix}x^2+x>x^2\\x^2+x< x^2+2x+1=\left(x+1\right)^2\end{matrix}\right.\)

\(\Rightarrow x^2< y^2< \left(x+1\right)^2\Rightarrow\) không tồn tại y nguyên thỏa mãn

- Với \(x\le-1\Rightarrow\left\{{}\begin{matrix}x^2+x=\left(x+1\right)^2-\left(x+1\right)\ge\left(x+1\right)^2\\x^2+x< x^2\end{matrix}\right.\)

\(\Rightarrow\left(x+1\right)^2\le y^2< x^2\Rightarrow y^2=\left(x+1\right)^2\)

\(\Rightarrow x^2+x=\left(x+1\right)^2\Rightarrow x+1=0\Rightarrow x=-1\Rightarrow y=0\)