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Question
If f (x) = |x|3, show that f ″(x) exists for all real x and find it.
Solution
Here, f(x) = |x|3 = x3
When, x > 0 |x| = x,
∴ f(x) = x3
f'(x) = 3x2, f'(x) = 6x …(1)
When, x < 0 |x| = - x
f(x) = |x|3 = (- x)3 = - x3
f'(x) = -3x2, f'(x) = - 6x …(2)
Thus,
`{(6x, if x>= 0),(-6x, if x < 0):}`
From (1) and (2),
f'(x) = 6|x|
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