Question

In the interval $\left[0,2\right],$ on which of the following function Lagrange's mean value theorem is not applicable?

• A. $f(x)=\left\{\begin{array}{cl}\frac{\sin x}{x} & x \neq 0 \\ 1 & x=0\end{array}\right.$
• B. $f(x)=\left\{\begin{array}{cc}1-x & x<1 \\ (1-x)^{2} & x \geq 1\end{array}\right.$
• C. $f\left(x\right)=x^{2}\left|x\right|$
• D. $f\left(x\right)=\left|e^{x} - 1\right|$

Answer 1

Answer: (B)

Note that LMVT is applicable in $\left[a , b\right]$ if $f\left(x\right)$ is continuous in $\left[a , b\right]$ and differentiable in $\left(a , b\right)$
Since $f(x)=\left\{\begin{array}{cl}1-x & x<1 \\ (1-x)^{2} & x \geq 1\end{array}\right.$ is non-differentiable at $x=1,$ hence LMVT cannot be applied to it in $\left[0,2\right]$