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Location of Roots when Compared to Two Constants 'k1' & 'k2'

If one of the...

Question

If one of the root of the qudratic polynomial $f (x) = a x^{2} + b x + c; a > 0$ is greater than $k_{1}$ and other root is less than $k_{2}$ . Then select the correct statement(s) for $k_{1} < k_{2}$ .

a \cdot f (k_{1}) < 0

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D > 0

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f (k_{1}) f (k_{2}) < 0

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a \cdot f (k_{2}) < 0

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Solution

The correct option is D $a \cdot f (k_{2}) < 0$
Let, $α, β$ be the roots of the quadratic polynomial $f (x) = a x^{2} + b x + c \forall a > 0$ where $α < k_{1} & β > k_{2}$ as shown:

From the graph, we can observe the following conditions:

$f (k_{1}) < 0 & f (k_{2}) < 0$
$∵ a > 0$
$\Rightarrow a . f (k_{1}) < 0 & a . f (k_{2}) < 0$
Also, $∵ f (k_{1}) < 0 & f (k_{2}) < 0$
$\Rightarrow f (k_{1}) \cdot f (k_{2}) > 0$

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