The value of c in Lagrange's theorem for the function $f\left(x\right)=\left\{\begin{array}{cc}x\cos \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}\right.$in the interval [-1, 1] is

f (x) is continuous in the interval [-1, 1], but f(x) is
not differentiable at x= 0. Hence, mean value theorem is
not applicable. So, no c can be found.