Question

For the given functions in $x\in [0,2],$ Rolle's theorem can be applicable on

• A. $\cos x$
• B. $x^3+x^2$
• C. $x^3-2x^2$
• D. $\tan x$

Answer 1

The correct option is C $x^3-2x^2$

For Rolle's theorem to be applicable function should be continuous in $[0,2]$ and differentiable in $(0,2)$ and $f\left(0\right)=f\left(2\right).$

For options $\left(a\right)$ and $\left(b\right):$ Both are continuous and differentiable in the given interval but $f\left(0\right)\ne f\left(2\right),$ so not applicable.

For $f\left(x\right)=\tan x$ is discontinuous at $x=\frac{\pi }{2}$

For $f\left(x\right)=x^3-2x^2=x^2(x-2),$ is continuous and differentiable everywhere also $f\left(0\right)=f\left(2\right),$ so rolle's theorem is applicable.