If a, b, c are in A.P., then for the equation 3ax² – 4bx + c = 0

$\begin{array}{l}\text{Let }\mathrm{f}\left(\mathrm{x}\right)={\mathrm{ax}}^3-2{\mathrm{bx}}^2+\mathrm{cx}\Rightarrow \mathrm{f}\left(0\right)=0\\ \mathrm{a},\mathrm{b},\mathrm{c} \mathrm{are} \mathrm{in} \mathrm{A}.\mathrm{P}\Rightarrow 2\mathrm{b}=\mathrm{a}+\mathrm{c} \therefore \mathrm{a}-2\mathrm{b}+\mathrm{c}=0\Rightarrow \mathrm{f}\left(1\right)\\ \Rightarrow \mathrm{f}\left(0\right)=0=\mathrm{f}\left(1\right)\\ \text{ According to ROLL's theorem f' }\left(\mathrm{x}\right)\text{=0 has atleast one root in between 0,1}\\ \therefore 3{\mathrm{ax}}^2-4\mathrm{bx}+\mathrm{c}=0 \mathrm{has} \mathrm{atleast} \mathrm{one} \mathrm{root} \mathrm{in} \left(0,1\right)\end{array}$