Question

Cho biểu thức: \(S=\left(1+\dfrac{a}{a^2+1}\right):\left(\dfrac{1}{a-1}-\dfrac{2a}{a^3+a-a^2-1}\right)\)với \(a\ne1\)

a) Rút gọn biểu thức S.

b) Tìm GTNN của biểu thức M = (a-1).S

Answer 1

\(a.S=\left(1+\dfrac{a}{a^2+1}\right):\left(\dfrac{1}{a-1}-\dfrac{2a}{a^3+a-a^2-1}\right)=\dfrac{a^2+a+1}{a^2+1}:\dfrac{a^2-2a+1}{\left(a^2+1\right)\left(a-1\right)}=\dfrac{a^2+a+1}{a^2+1}.\dfrac{a^2+1}{a-1}=\dfrac{a^2+a+1}{a-1}\)

\(b.M=\left(a-1\right).S=a^2+a+1=a^2+2.\dfrac{1}{2}a+\dfrac{1}{4}+1-\dfrac{1}{4}=\left(a+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\)

\(\Rightarrow M_{MIN}=\dfrac{3}{4}."="\Leftrightarrow a=-\dfrac{1}{2}\)