Question

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


Which of the following options is/are true for the graph?

• A. $9a-3b+c<0$
• B. $4a-2b+c<0$
• C. $a-b+c>0$
• D. $4a-6b+9c>0$

Answer 1

Answer: (C), (D)

$4a-6b+9c>0$

Given: $f\left(x\right)=a{x}^2+bx+c$

From the given graph,
$f\left(x\right)>0,\forall x\in R$
Now,
$f(-1)>0$
$a(-1{)}^2+b(-1)+c>0$
$\Rightarrow a-b+c>0$

Similarly,
$f(-2)>0$
$a(-2{)}^2+b(-2)+c>0$
$\Rightarrow 4a-2b+c>0$

$f(-3)>0$
$a(-3{)}^2+b(-3)+c>0$
$\Rightarrow 9a-3b+c>0$

$f(-\frac{2}{3})>0$
$a{(-\frac{2}{3})}^2+b(-\frac{2}{3})+c>0$
$\Rightarrow \frac{4a}{9}-\frac{2b}{3}+c>0$
$\Rightarrow 4a-6b+9c>0$