One or more options can be correct:

The condition for the roots of the quadratic equation $f (x) = a x^{2} + b x + c$ to be less than a real value $x_{0}$ is/are

$f (x_{0}) < 0$

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$D \geq 0$

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$f (x_{0}) > 0$

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$x_{0} > \frac{- b}{2 a}$

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Solution

The correct option is D
$x_{0} > \frac{- b}{2 a}$

Conditions for both roots to be less than a real value $x_{0}$ , can be visualized through graph

Here, $a = 1 (> 0), b = - 2 m, c = m^{2} - 1$

Observe that
$(i) f (x_{0}) > 0$

$(i i) x_{0} > \frac{- b}{2 a}$

Also for real roots to exist
$(i i i) b^{2} \geq 4 a c$

Hence, $A, C, D$ are right options.

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