If a,b,c∈R and a+b+c=0, then the quadratic equation 3ax2+2bx+c=0 has

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If $a, b, c \in R$ and a+b+c=0, then the quadratic equation $3 a x^{2} + 2 b x + c = 0$ has

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at least one root in [1,2]

none of these.

at least one root in [0, 1]

at least one root in [3/2, 2]

answer is A.

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Detailed Solution

Let $f (x) = a x^{3} + b x^{2} + c x$ Note that f is continuous and derivable on R. Also $f (0) = 0$ and least one $α \in (0, 1)$ such that $f^{'} (α) = 0 \Rightarrow 3 a α^{2} + 2 b α + c = 0$ Thus, $3 a x^{2} + 2 b x + c = 0$ has at least one root in [0,1].