# Answer to Question #15614 in Calculus for hsd

Question #15614

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.

Determine all x (is a real number) at which f(x) = |x - 2| is differentiable and compute f'(x) if possible.

Expert's answer

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.

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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