Multiple choice

If $a + b + c = 0$, then the equation $3a{x}^{2} + 2bx + c = 0$ has, in the interval $\left(0, 1\right)$

  1. Atleast one root

  2. Atmost one root

  3. No root

  4. None of these
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A Correct answer
AI explanation

Consider the polynomial f(x) equals 3ax cubed plus bx squared plus cx, whose derivative is 9ax squared plus 2bx plus c, matching the given equation. Since a plus b plus c equals 0, we have f(1) equals 3a plus b plus c, which is greater than f(0) equals 0 because 2a plus (a plus b plus c) equals 2a. By Rolle's theorem, since the derivative equals the given quadratic, it must have at least one root between 0 and 1. The equation has at least one root in the interval.

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