As, polynomial function is continuous everywhere in its domain. So, f[g(x)] will be continuous everywhere at x<0,0<x<1 and x>1. We will check the behaviour of fog(x) only at boundary points which is x=0 and x=1. At x=0,lim−((x3+2)=2
lim
x→0+
x6=0 Clearly, LHL≠RHL at x=0 So, f∘g(x) is discontinuous at x=0. ⇒fog(x) is not differentiable at x=0 At x=1,
lim
x→1−
x6=1
lim
x→1+
(3x−2)2=1. Also f(1)=1 fog (x) is continuous at x=1 Derivative test at x=1, LHD=
lim
h→0
‌
f(1)−f(1−h)
h
=
lim
h→0
‌
1−(1−h)6
h
=
lim
h→0
6(1−h)5=6 RHD=
lim
h→0
‌
f(1+h)−f(1)
h
=
lim
h→0
‌
[3(1+h)−2]2−1
h
=
lim
h→0
2[3(1+h)−2]⋅3=6
∴f∘g(x) is continuous and differentiable at x=1. ∴fog(x) is discontinuous and non-differentiable at x=0. So, number of points of non-differentiability of f∘g(x) is 1 .