Question

tìm GTNN của biểu thức M = \(\sqrt{x^2+x+1}+\sqrt{x^2-x+1}\)

Answer 1

\(M=\sqrt{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}}+\sqrt{\left(x-\frac{1}{2}\right)^2+\frac{3}{4}}\)

\(=\sqrt{\left(x+\frac{1}{2}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2}+\sqrt{\left(\frac{1}{2}-x\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2}\)

\(\Rightarrow M\ge\sqrt{\left(x+\frac{1}{2}+\frac{1}{2}-x\right)^2+\left(\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}\right)^2}=2\)

\(\Rightarrow M_{min}=2\) khi \(x+\frac{1}{2}=\frac{1}{2}-x\Rightarrow x=0\)