Mathematics - Quadratic Equation Question with Solution | TestHub

Let $f(x)=(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)$.

According to the given condition, we have:

$$\begin{array}{ll}& f(0)f(-1)<0\\ \text{ i.e. }& (c+a-2b)(2a-b-c)<0\\ \text{ i.e. }& (c+a-2b)(a-b+a-c)<0\\ \text{ i.e. }& c+a-2b<0\\ \text{ i.e. }& b>\frac{a+c}{2}\end{array}$$

$\Rightarrow b$ cannot be the G.M. of $a,c$, since G.M < A.M. always.