If $a+b+c=0$, then the quadratic equation $3a{ x }^{ 2 }+2bx+c=0$ has atleast one root in

To find the interval containing a root, we use Rolle's Theorem by identifying a function whose derivative is proportional to the given quadratic expression. Let $f(x) = ax^3 + bx^2 + cx$, making its derivative $f'(x) = 3ax^2 + 2bx + c$. We evaluate the function at $x = 0$ and $x = 1$, yielding $f(0) = 0$ and $f(1) = a + b + c$. Since the problem states that $a + b + c = 0$, we have $f(1) = 0$. Because $f(0) = f(1)$, Rolle's Theorem guarantees that the derivative $f'(x)$ equals zero for at least one value in the interval between 0 and 1. Therefore, the quadratic equation has at least one real root in $(0, 1)$.

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