Question

\(CMR:x\left(x-1\right)+y\left(y-1\right)+z\left(z-1\right)\le\frac{4}{3}\)

\(CMR:-1\le x+y+z\le4\)

Answer 1

ta co  3(x²+y²+z²)-3(x+y+z)<=4

de dang chung minh bdt 3(x²+y²+z²)>=(x+y+z)²

ap dung bat dang thuc ta co

3(x²+y²+z²)-(x+y+z)>=(x+y+z)²-3(x+y+z)

=>(x+y+z)²-3(x+y+z)-4<=0

=>(x+y+z+1)(x+y+z-4)<=0

=>-1<=x+y+z=<4 (dpcm)

Answer 2

ta co  3(x²+y²+z²)-3(x+y+z)<=4

de dang chung minh bdt 3(x²+y²+z²)>=(x+y+z)²

ap dung bat dang thuc ta co

3(x²+y²+z²)-(x+y+z)>=(x+y+z)²-3(x+y+z)

=>(x+y+z)²-3(x+y+z)-4<=0

=>(x+y+z+1)(x+y+z-4)<=0

=>-1<=x+y+z=<4 (dpcm)