Question

Tìm GTNN của biểu thức:

\(N=\dfrac{3x^2-4x}{x^2+1}\)

\(P=\dfrac{2x+1}{x^2+2}\)

Answer 1

\(N=\frac{3x^2-4x}{x^2+1}=\frac{4x^2-4x+1-\left(x^2+1\right)}{x^2+1}=\frac{\left(2x-1\right)^2}{x^2+1}-1\ge-1\forall x\)

Dấu "=" xảy ra khi \(2x-1=0\Rightarrow x=\frac{1}{2}\)

Vậy \(MinN=-1\Leftrightarrow x=\frac{1}{2}\)

\(P=\frac{2x+1}{x^2+2}=\frac{4x+2}{2x^2+4}=\frac{x^2+4x+4-\left(x^2+2\right)}{2x^2+4}=\frac{\left(x+2\right)^2}{2x^2+4}-\frac{1}{2}\ge-\frac{1}{2}\forall x\)

Dấu "=" xảy ra khi: \(x+2=0\Rightarrow x=-2\)

Vậy \(MinP=-\frac{1}{2}\Leftrightarrow x=-2\)