AP EAMCET · Maths · Continuity and Differentiability
Match the functions in Column I with their properties in Column II. In the following \([\mathrm{x}]\) denotes the greatest integer less than or equal to x
| Column I | Column II | ||
|---|---|---|---|
| A) | \(x|x|\) | I. | Strictly increasing and continuous in \((-1,1)\) |
| B) | \(\sqrt{|x|}\) | II. | Continuous but not differentiable in \((-1,1)\) |
| C) | \(x+[x]\) | III. | Differentiable in \((-1,1)\) |
| D) | \(|x-1|+|x+1|+|x|\) | IV. | Differentiable in \((-1,0) \cup(0,1)\) |
| V. | Strictly increasing and not differentiable in \((-1,1)\) |
The correct match is
- A A-III, B-V, C-II, D-I
- B A-II, B-III, C-I, D-V
- C A-I, B-II, C-V, D-IV
- D A-IV, B-I, C-V, D-III
Answer & Solution
Correct Answer
(C) A-I, B-II, C-V, D-IV
Step-by-step Solution
Detailed explanation
A) For \(f(x) = x|x|\): \(f(x) = \begin{cases} x^2 & \text{if } x \ge 0 \\ -x^2 & \text{if } x Continuous in \((-1,1)\). \(f'(x) = \begin{cases} 2x & \text{if } x > 0 \\ -2x & \text{if } x \(f'(x) = 2|x| \ge 0\) for all \(x \in (-1,1)\) and \(f'(x)=0\) only at…
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