Question

For which of the following functions Rolle's theorem is not applicable :

• A. $f\left(x\right)=\sqrt{4-x^2}$ in $[-2,2]$
• B. $f\left(x\right)=\left[x\right]$ in $[-1,1]$
  $[.]$ denotes the greatest integer function
• C. $f\left(x\right)=x^2+3x-4$ in $[-4,1]$
• D. $f\left(x\right)=\cos 2x$ in $[0,\pi ]$

Answer 1

The correct option is B $f\left(x\right)=\left[x\right]$ in $[-1,1]$

$[.]$ denotes the greatest integer function
Rolle's Theorem can be applied when the function is continuous and differentiable in given interval. Also, its values at the end points has to be same.
Checking by graphs:
$f\left(x\right)=\sqrt{4-x^2}$ in $[-2,2]$
$f\left(x\right)=\left[x\right]$ in $[-1,1]$

$f\left(x\right)=x^2+3x-4$ in $[-4,1]$
$f\left(x\right)=\cos 2x$ in $[0,\pi ]$Hence only $f\left(x\right)=\left[x\right]$ is discontinuous in its given domain, and hence it is the only function which does not satisfy the conditions of Rolles theorem.