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Lagrange's mean value theorem is not applicable in $[0,1]$ to the function

MathematicsApplication of DerivativesAP EAMCETAP EAMCET 2017 (24 Apr Shift 2)

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A $f(x)=\left\{\begin{array}{l}\frac{1}{2}-x, x < \frac{1}{2} \\ \left(\frac{1}{2}-x\right)^2, x \geq \frac{1}{2}\end{array}\right.$
B $f(x)=\left\{\begin{array}{c}\frac{\sin x}{x}, x \neq 0 \\ 1, x=0\end{array}\right.$
C $f(x)=x|x|$
D $f(x)=|x|$