Question

Cmr:
1, \(f\left(x\right)=25x^2-20x+\dfrac{9}{2}>0\)
2, \(f\left(x\right)=4x^2-28x+50>0\)
3, \(f\left(x\right)=-16x^2+72x-82< 0\)
4, \(f\left(x\right)=9x^2+66x-122< 0\)
5, \(f\left(x;y\right)=4x^2+9y^2-12x+6y+11>0\)
6, \(f\left(x;y\right)=13x^2+y^2+4xy-34x-2y+27>0\)

Answer 1

1. \(f\left(x\right)=25x^2-20x+\dfrac{9}{2}\)

=>\(f\left(x\right)=25x^2-20x+4+\dfrac{1}{2}\)

=> \(f\left(x\right)=(25x^2-20x+4)+\dfrac{1}{2}\)

=> \(f\left(x\right)=(5x-2)^2+\dfrac{1}{2}\)

Ta thấy: \((5x-2)^2\ge0\)

=>\(f\left(x\right)=(5x-2)^2+\dfrac{1}{2}\ge\dfrac{1}{2}>0\)(đpcm)

2. \(f\left(x\right)=4x^2-28x+50\)

=> \(f\left(x\right)=(4x^2-28x+49)+1\)

=> \(f\left(x\right)=(2x-7)^2+1\)

Ta thấy: \((2x-7)^2\ge0\)

=> \(f\left(x\right)=(2x-7)^2+1\ge1>0\) (đpcm)

3. \(f\left(x\right)=-16x^2+72x-82\)

=> \(f\left(x\right)=-(16x^2-72x+82)\)

=> \(f\left(x\right)=-(16x^2-72x+81+1)\)

=> \(f\left(x\right)=-[(4x-9)^2+1]\)

Ta thấy: \((4x-9)^2\ge0\)

=> \((4x-9)^2+1\ge1>0\)

=> \(f\left(x\right)=-[(4x-9)^2+1]< 0\)

5. \(f\left(x;y\right)=4x^2+9y^2-12x+6y+11\)

=> \(f\left(x;y\right)=4x^2+9y^2-12x+6y+9+1+1\)

=> \(f\left(x;y\right)=(4x^2-12x+9)+(9y^2+6y+1)+1\)

=> \(f\left(x;y\right)=(2x-3)^2+(3y+1)^2+1\)

Ta thấy: \((2x-3)^2\ge0\)

\((3y+1)^2\ge0\)

=> \(f\left(x;y\right)=(2x-3)^2+(3y+1)^2+1\) \(\ge1>0\) (đpcm)