Question

In which of the following functions, Rolle's theorem is applicable?

• A. $f\left(x\right)=|x|$ in $-2\le x\le 2$
• B. $f\left(x\right)=tanx$ in $0\le x\le \pi$
• C. $f\left(x\right)=1+(x-2{)}^{\frac{2}{3}}$ in $1\le x\le 3$
• D. $f\left(x\right)=x(x-2{)}^2$ in $0\le x\le 2$

Answer 1

The correct option is D $f\left(x\right)=x(x-2{)}^2$ in $0\le x\le 2$

$f\left(x\right)=|x|$ not differentiable at $x=0$
$f\left(x\right)=tanx$ discontinuous at $x=\frac{\pi }{2}$
$f\left(x\right)=1+(x-2{)}^{\frac{3}{2}}$ not differentiable at $x=2$
$f\left(x\right)=x(x-2{)}^2$ polynomial.
$\therefore$ Differential $forallx\in R$
Hence, rolle's theorem is applicable.