Question

[ngoc bich]

Tìm GTNN của \(E=\frac{x}{\left(x+2010\right)^2}\)

Giúp với!!!!

Answer 1

[ST]

Đặt \(t=\frac{1}{x+2010}\Rightarrow x=\frac{1}{t}-2010\)

Ta có: \(E=x\cdot\frac{1}{\left(x+2010\right)^2}=\left(\frac{1}{t}-2010\right)t^2=t-2010t^2\)

\(=-2010\left(t^2-t\cdot\frac{1}{2010}\right)=-2010\left(t^2-2t\cdot\frac{1}{4020}+\frac{1}{4020^2}\right)+\frac{1}{8040}\)

\(=-2010\left(t-\frac{1}{4020}\right)^2+\frac{1}{8040}\le\frac{1}{8040}\)

Dấu "=" xảy ra <=> \(t=\frac{1}{4020}\Leftrightarrow x=2010\)