Question

If one of the root of the qudratic polynomial $f\left(x\right)=a{x}^2+bx+c;a>0$ is greater than $k_1$ and other root is less than $k_2$. Then select the correct statement for $k_1<k_2$.

• A. $a\cdot f\left(k_1\right)<0$
• B. $f\left(k_1\right)f\left(k_2\right)<0$
• C. $a\cdot f\left(k_2\right)>0$
• D. $D<0$

Answer 1

The correct option is A $a\cdot f\left(k_1\right)<0$

Since, the given quadratic polynomial has real roots $\Rightarrow D>0$
Let $\alpha ,\beta$ be the roots of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c,$ where $\alpha >k_1\&\beta <k_2,$ as shown:


From the graph, we get:

$f\left(k_1\right)<0\&f\left(k_2\right)<0$

Also $a>0$ [Given]

$\Rightarrow a.f\left(k_1\right)<0$ and $a.f\left(k_2\right)<0$