Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | \(f(x) = |x|\) | (I) | Not differentiable at \(x=-2\) only |
| (B) | \(f(x) = |x+2|\) | (II) | Not differentiable at \(x = 0\) only |
| (C) | \(f(x) = |x^2-4|\) | (III) | Not differentiable at \(x = 2\) only |
| (D) | \(f(x)=|x-2|\) | (IV) | Not differentiable at \(x = 2,-2\) only |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(II), (B)-(I), (C)-(IV), (D)-(III)
A function |g(x)| with g a polynomial has a corner (left and right derivatives differ) at a point where g(x) = 0 and \(g'(x) \ne 0\), so it is not differentiable there; it is differentiable everywhere else.
(A) \(f(x) = |x|\)
This matches (II) Not differentiable at x = 0 only. At 0 the left derivative is −1 and the right derivative is 1.
(B) \(f(x) = |x+2|\)
This matches (I) Not differentiable at x = −2 only. Here x + 2 = 0 at x = −2, where the left derivative is −1 and the right derivative is 1.
(C) \(f(x) = |x^2-4|\)
This matches (IV) Not differentiable at x = 2, −2 only. Here \(x^2 - 4 = 0\) at \(x = \pm 2\), and its derivative 2x is not 0 at either point. At x = 2, for example:
\(\text{left derivative} = -2x = -4\)
\(\text{right derivative} = 2x = 4\)
(D) \(f(x)=|x-2|\)
This matches (III) Not differentiable at x = 2 only. At 2 the left derivative is −1 and the right derivative is 1.
So the matching is (A)-(II), (B)-(I), (C)-(IV), (D)-(III).