Question

Tìm GTLN của biểu thức

\(E=\dfrac{\left|2x-1\right|+5}{\left|2x-1\right|+3}\)

Answer 1

\(E=\dfrac{\left|2x-1\right|+5}{\left|2x-1\right|+3}=\dfrac{\left|2x-1\right|+3+2}{\left|2x-1\right|+3}=1+\dfrac{2}{\left|2x-1\right|+3}\)

Do \(\dfrac{2}{\left|2x-1\right|+3}\le\dfrac{2}{3}\forall x\)

\(\Rightarrow1+\dfrac{2}{\left|2x-1\right|+3}\le\dfrac{5}{3}\forall x\)

\(\Rightarrow E\le\dfrac{5}{3}\forall x\)

Dấu " = " xảy ra

\(\Leftrightarrow\dfrac{2}{\left|2x-1\right|+3}=\dfrac{2}{3}\)

\(\Leftrightarrow\left|2x-1\right|+3=3\)

\(\Leftrightarrow\left|2x-1\right|=0\)

\(\Leftrightarrow2x-1=0\Leftrightarrow2x=1\)

\(\Leftrightarrow x=\dfrac{1}{2}\)

Vậy Max E là : \(\dfrac{5}{3}\Leftrightarrow x=\dfrac{1}{2}\)