A function with absolute value ∣g(x)∣ is not differentiable where g(x)=0, as this creates a sharp corner.
For f(x)=∣x∣:
The expression inside is x.
Setting x=0 gives the point where the function is not differentiable.
Not differentiable at x=0 only.
(A) matches with (II)
For f(x)=∣x+2∣:
The expression inside is x+2.
x+2=0
x=−2
Not differentiable at x=−2 only.
(B) matches with (I)
For f(x)=∣x2−4∣:
The expression inside is x2−4.
x2−4=0
x2=4
x=±2
Not differentiable at x=2,−2 only.
(C) matches with (IV)
For f(x)=∣x−2∣:
The expression inside is x−2.
x−2=0
x=2
Not differentiable at x=2 only.
(D) matches with (III)
Final matching:
(A) → (II)
(B) → (I)
(C) → (IV)
(D) → (III)