Limits Continuity and Differentiability

(I) f(x)={

‌ 1 2 −x,‌	x<‌ 1 2
(‌ 1 2 −x)2	x≥‌ 1 2

f(x) is not differentiable at x=‌

1

2

LMVT not applicable
(II) f(x)=|3x−1|
f(x) is not differentiable at x=‌

1

3

∴ LMVT is not applicable.
(III) f(x)=x|x| it is continuous and differentiable in ( 0,1 )
= LMVT applicable.
(V) f(x)=|x| it is continuous and differentiable in ( 0,1 )
∴ LMVT is applicable.