Question

cho \(f\left(x\right)=ãx^2+bx+c\)

CMR \(f\left(2\right).f\left(-3\right)\le0\) nếu \(13a+b+2c=0\)

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Answer 1

Ta có:

\(f\left(x\right)=ax^2+bx+c\)

\(\Rightarrow f\left(2\right).f\left(-3\right)=\left(4a-2b+c\right)\left(9a+3b+c\right)\)

\(=\left(4a-2b+c\right)\left[13a+b+2c-\left(4a-2b+c\right)\right]\)

Mà \(13a+b+c=0\)

\(\Rightarrow f\left(2\right).f\left(-3\right)=-\left[\left(4a-2b+c\right)^2\right]\)

Ta có \(\left(4a-2b+c\right)^2\ge0\Rightarrow-\left[\left(4a-2b+c\right)^2\right]\le0\)

 Vậy nếu \(13a+b+2c=0\)\(\Rightarrow f\left(2\right).f\left(3\right)\le0\) (Đpcm)