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Question
The set of all points where the function f(x) = x + |x| is differentiable, is ______.
Options
(0, ∞)
(–∞, 0)
(–∞, 0) ∪ (0, ∞)
(–∞, ∞)
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Solution 1
The set of all points where the function f(x) = x + |x| is differentiable, is (–∞, 0) ∪ (0, ∞).
Explanation:
f(x) = x + |x| = `{{:(2x",", x ≥ 0),(0",", x < 0):}`
There is a sharp corner at x = 0, so f(x) is not differentiable at x = 0.
Solution 2
The set of all points where the function f(x) = x + |x| is differentiable, is (–∞, 0) ∪ (0, ∞).
Explanation:
Lf' (0) = 0 and Rf' (0) = 2 ; so, the function is not differentiable at x = 0
For x ≥ 0, f(x) = 2x (linear function) and when x < 0, f(x) = 0 (constant function)
Hence f(x) is differentiable when x ∈ (–∞, 0) ∪ (0, ∞).
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