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If $2 a + 3 b + 6 c = 0,$ the at least one root of the equation $a x^{2} + b x + c = 0$ lies in the interval

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Correct option is A

Consider the function

$f (x) = \frac{a x^{3}}{3} + \frac{b x^{2}}{2} + c x$

We find that $f (0) = 0$ and,

$f (1) = \frac{a}{3} + \frac{b}{2} + c = \frac{2 a + 3 b + 6 c}{6} = \frac{0}{6} = 0 [∵ 2 a + 3 b + 6 c = 0]$

Therefore, 0 and 1 are roots of the polynomial $f (x) .$

Consequently, there exists at least one root of the polynomial

$f^{'} (x) = a x^{2} + b x + c$ lying between 0 and 1.

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If 2a+3b+6c=0, the at least one root of the equation ax2+bx+c=0 lies in the interval