Question

Let $g\left(x\right)$ is a continuous and differentiable function and $f\left(x\right)=x\cdot g\left(x\right)$ has a positive root at $x=\alpha ,$ then one root of $f^{\prime }\left(x\right)=0$ can be

• A. $-\frac{\alpha }{2}$
• B. $\frac{\alpha }{2}$
• C. $\frac{\alpha }{3}$
• D. $\frac{3\alpha }{2}$

Answer 1

Answer: (B), (C)

$\frac{\alpha }{3}$

Given : $f\left(x\right)=x\cdot g\left(x\right)$ is continuous and differentiable.
Clearly, $f\left(0\right)=0,f\left(\alpha \right)=0$

We know, between two roots of $f\left(x\right)=0,$ there must be atleast one $c\in (0,\alpha )$ where $f^{\prime }\left(c\right)=0$
So, $f^{\prime }\left(x\right)=0$ has atleast one root which lies in $(0,\alpha )$