Question

If rolle's theorem is applicable on $f\left(x\right)=x^{\alpha }\tan x$ in $[-\frac{\pi }{4},\frac{\pi }{4}],$ then the value of $\alpha +1$ can be

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

Given : $f\left(x\right)=x^{\alpha }\tan x$
clearly, for $\alpha <0;f\left(x\right)$ will be discontinuous at $x=0$

Now for $\alpha \ge 0:f\left(x\right)$ is continuous and differentiable in the given interval.
and $f\left(\frac{\pi }{4}\right)={\left(\frac{\pi }{4}\right)}^{\alpha },$
$f(-\frac{\pi }{4})={(-\frac{\pi }{4})}^{\alpha }\cdot (-1)=(-1{)}^{\alpha +1}\cdot {\left(\frac{\pi }{4}\right)}^{\alpha }$
for $f\left(\frac{\pi }{4}\right)=f(-\frac{\pi }{4})$
$\Rightarrow \alpha +1\in$ even and $\alpha \ne 0$