Question

CMR:

a, với mọi x ≠ ±1 thì A= \(\frac{x+2}{x-1}\)(\(\frac{x^3}{2x+2}\)+1) - \(\frac{8x+7}{2x^2-2}\) luôn có giá trị dương

b, với mọi x ≠ 0 và x ≠ -3 thì B= \(\frac{1-x^2}{x}\)(\(\frac{x^2}{x+3}\) - 1) + \(\frac{3x^2-14x+3}{x^2+3x}\) luôn nhận giá trị âm

Giúp mk nha mng, thx!!

Answer 1

\(A=\frac{x+2}{x-1}\left(\frac{x^3+2x+2}{2\left(x+1\right)}\right)-\frac{8x+7}{2\left(x^2-1\right)}\)

\(=\frac{\left(x+2\right)\left(x^3+2x+2\right)-8x-7}{2\left(x-1\right)\left(x+1\right)}\)

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

Answer 2

b, \(B=\frac{1-x^2}{x}.\frac{x^2-\left(x+3\right)}{x+3}+\frac{3x^2-14x+3}{x\left(x+3\right)}\)

\(=\frac{-x^4+x^3+7x^2-15x}{x\left(x+3\right)}\)

\(=\frac{-x^2\left(x+3\right)+4x\left(x+3\right)-5\left(x+3\right)}{x+3}\)

\(=\frac{\left(x+3\right)\left(-x^2+4x-5\right)}{x+3}\)

\(=-\left(x^2-4x+5\right)\)

\(=-\left(x^2-4x+4+1\right)\)

\(=-[\left(x-2\right)^2+1]\)

=> B luôn nhận giá trị âm do (x-2)^2 + 1 >0 với mọi x thuộc R