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Question
If f(x) = |x|3, show that f"(x) exists for all real x and find it.
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Solution
Case I: When x ≥ 0
Here, f(x) = |x|3 = x3
∴ f'(x) = 3x2
f''(x) = 6x
Case II: When, x < 0
Here, f(x) = (−x)3 = −x3
∴ f'(x) = −3x2
f"(x) = −6x
Thus, f"(x) = `{(6x", if" x>= 0),(-6x", if" x < 0):}`
Hence f"(x) = 6|x|
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