Question
If $$a + b + c = 0,$$ then the quadratic equation $$3a{x^2}+ 2bx + c = 0$$ has
A.
at least one root in $$\left[ {0, 1} \right]$$
B.
one root in $$\left[ {2, 3} \right]$$ and the other in $$\left[ { - 2, - 1} \right]$$
C.
imaginary roots
D.
none of these
Answer :
at least one root in $$\left[ {0, 1} \right]$$
Solution :
Consider the function $$f\left( x \right) = a{x^3} + b{x^2} + cx$$ on $$\left[ {0, 1} \right]$$ then being a polynomial. It is continuous on $$\left[ {0, 1} \right],$$ differentiable on $$\left( {0, 1} \right)$$ and
$$f\left( 0 \right) = f\left( 1 \right) = 0$$ [as given $$a + b + c = 0$$ ]
$$\therefore $$ By Rolle's theorem $$\exists \,x \in \left( {0,1} \right)$$ such that
$$f'\left( x \right) = 0 \Rightarrow 3a{x^2} + 2bx + c = 0$$
Thus equation $$3a{x^2} + 2bx + c = 0$$ has at least one root in [0, 1].