Mathematics - Quadratic Equation Question with Solution | TestHub

Let $f(x)=\left(x-a_1\right)\left(x-a_2\right)\dots \dots \dots \dots ..(x-an)$ where $a_1,a_2\dots \dots ..a_{n\in R}$ take log both sides and differentiate. Then

$$\frac{f^{\prime }(x)}{f(x)}=\frac{1}{x-a_1}+\frac{1}{x-a_2}+\dots \dots \dots .+\frac{1}{x-a_n}$$

Again diff w.r.t. ' x '

$$\begin{array}{rlrl}& \frac{f{f}^{\prime \prime }-{\left(f^{\prime }\right)}^2}{f^2}=-\left[\frac{1}{{\left(x-a_1\right)}^2}+\frac{1}{{\left(x-a_2\right)}^2}+\dots \dots \frac{1}{{\left(x-a_n\right)}^2}\right]& <0\forall x\in R& \Rightarrow f{f}^{\prime \prime }-{\left(f^{\prime }\right)}^2=0\text{ has no real root }\end{array}$$