Question

If a + b + c = 0, then the quadratic equation $3a{x}^2+2bx+c=0$ has at least one root in (0, 1)

Answer 1

Let F(x) = $\int (3a{x}^2+2bx+c)dx=a{x}^3+b{x}^2+cx$
$\therefore$ F(0) = 0, F(1) = a + b + c = 0 by given condition.
Since both F(0) = 0 and F(1) = 0,
there will exist some value of x between 0 and 1 for which F' (x) = 0 by Rolle's theorem.
where F' (x) = $3a{x}^2$ + 2bx + c.