Answer to Question #15614 in Calculus for hsd

Question

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.

Answer 1

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.