Question

Cho biểu thức A = \(\dfrac{x}{x+1}-\dfrac{3-3x}{x^2-x+1}+\dfrac{x+4}{x^3+1}\left(x\ne-1\right)\)

a, Rút gọn biểu thức A

b, CMR \(A>0\forall x\ne-1\)

c, Với x > 0. Tính GTLN của A

Answer 1

a: \(=\dfrac{x^3-x^2+x+3\left(x^2-1\right)+x+4}{\left(x+1\right)\left(x^2-x+1\right)}\)

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

\(=\dfrac{\left(x+1\right)\left(x^2-x+1\right)+2x\left(x+1\right)}{\left(x+1\right)\left(x^2-x+1\right)}\)

\(=\dfrac{x^2+x+1}{x^2-x+1}\)

b: \(x^2+x+1=\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}>0\)

\(x^2-x+1=\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}>0\)

=>A>0 với mọi x<>-1