for lagrange's theorem,
1) it should be continous at [-2,0]
f(x) = |x+2| =-(x+2 ) x< -2
(x+2) -2<= x< -1
(x+2) -1< x < 0
(x+2) x>0
so,for continouty check limit at -2 _{lim x ->-2-} -(x+2) =0
rhs at -2------ lim x-> -2^{+ }(x+2) =0
and f(-2) =0
hence,continous at x=-2
similarly,it is continous at -1 and 0
2) f(x) should be differentiable at x= ]-2,0[
for checking diferentibilty
lim x-> c^{-} f(x) -f(c)/x-c =lim x->c^{+ }f(x) -f(c)/x-c
which is same as f(x) for lhs and rhs of -1 is same.
it should follow lagranje