If a + b + c = 0,then how many roots does the | vedclass

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(A) Let $f(x) = ax^3 + bx^2 + cx$. Since $f(x)$ is a polynomial,it is continuous on $[0, 1]$ and differentiable on $(0, 1)$. Calculate the values at t

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At least one root in $(0, 1)$

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(A) Let $f(x) = ax^3 + bx^2 + cx$. Since $f(x)$ is a polynomial,it is continuous on $[0, 1]$ and differentiable on $(0, 1)$. Calculate the values at the endpoints: $f(0) = a(0)^3 + b(0)^2 + c(0) = 0$. $f(1) = a(1)^3 + b(1)^2 + c(1) = a + b + c$. Given that $a + b + c = 0$,we have $f(1) = 0$. Since $f(0) = f(1) = 0$,by Rolle's Theorem,there exists at least one $c_0 \in (0, 1)$ such that $f'(c_0) = 0$. $f'(x) = 3ax^2 + 2bx + c$. Thus,the equation $3ax^2 + 2bx + c = 0$ has at least one root in the interval $(0, 1)$.

If $a + b + c = 0$,then how many roots does the equation $3ax^2 + 2bx + c = 0$ have in the interval $(0, 1)$?

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