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
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
for roll's theorem,there is one more condition,
f(a) = f(b) should be satisfied
whereas cauchy is applied on two functions defined on same domain
1) f and g should be continous in [a,b]
2) f and g should be derivable in ]a,b[
3) g'(x) !=0 for all x in ]a,b[
u r missing one very important point in all of these...continuity and differentiability is fine..but to say whether lagranges theorem will be applicble or not u need to find a c(mean value) belonging to (a,b)..isnt it?
and same with rolles theorem and cauchy..
X->YZ , Y->XZ , ...