Question

Tìm GTLN, GTNN của: \(A=\dfrac{2x+1}{x^2+2}\)

Answer 1

\(A=\frac{\left(x^2+2\right)-\left(x^2-2x+1\right)}{x^2+2}=1-\frac{\left(x-1\right)^2}{x^2+2}\le1\forall x\)

Dấu "=" xảy ra khi: \(x-1=0\Rightarrow x=1\)

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

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

Vậy \(MaxA=1\Leftrightarrow x=1,MinA=-\frac{1}{2}\Leftrightarrow x=-2\)