Rolle’s Theorem states that, for a function $f$ if:
- $f$ is continuous in range $[a,b]$
- $f$ is differentiable in range $(a,b)$
- $f(a) = f(b)$
then, $f’(x)=0, for some$ $a\leq x \leq b$
Here, $ax^2 + bx + c = 0$ means that we have been given $f’(x)$
$\therefore After\:integrating\: f’(x),\: we\: get\: f(x) $
$\therefore f(x)= \frac{ax^3}{3}+\frac{bx^2}{2} + cx + d$
$Now,$
$f(0) = d$
$f(1) = \frac{a}{3}+\frac{b}{2} + c + d = \frac{2a+3b+6c+6d}{6} = \frac{6d}{d} = d\;(\because 2a+3b+6c=0)$
$So,f(0)=f(1) \:means\:that\:there\:is\:atleast\:one\:root\:between\:0\:and\:1$
$So, option\: A\:is\: correct\: choice.$