Recall the definition of the absolute value, |x| = x; if x ≥ 0; -x if x ≤ 0:
Determine all x (is a real number) at which f(x) = |x - 2| is differentiable and compute f'(x) if possible.
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Expert's answer
2012-10-02T10:37:26-0400
The function f(x) = |x - 2| is defined by f(x) = 2-x, for x<2 and f(x) = x-2, for x>=2
Hence f'(x)=(2-x)' = -1, for x<2 f'(x)=(x-2)' = +1, for x>=2
Hence f is differentiable at all points x<>2. For x=2 the function is not differentiable, since the limits of f' to x from the left and from the right are distinct.
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