Question

The graph of quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$ is shown below


Which of the following statement(s) is/are correct?

• A. $f\left(x\right)>0;\forall x>\beta$
• B. $ac>0$
• C. $\frac{c}{a}<-1$
• D. $\frac{c}{a}(\alpha -\beta )>2$

Answer 1

Answer: (C), (D)

$\frac{c}{a}(\alpha -\beta )>2$


$\alpha <-1,\beta >1$
Graph is downward, so
$a<0$
$f\left(0\right)=c$, so from the graph,
$c>0$
So,
$ac<0$

Now,
$-\alpha >1,\beta >1$
$\Rightarrow -\alpha \beta >1$
$\Rightarrow \alpha \beta <-1$
$\Rightarrow \frac{c}{a}<-1$

Now,
$\frac{c}{a}(\alpha -\beta )=(-\alpha \beta )(\beta -\alpha )$
We know that,
$(\beta -\alpha )>2(\because -\alpha >1,\beta >1)$
$\alpha \beta <-1\Rightarrow -\alpha \beta >1$
$\therefore \frac{c}{a}(\alpha -\beta )>2$