Question: Show that the function f ( x ) = | x 5| is not differentiable at5. We have f ( x ) = | x

Show that the functionf(x) = |x5|

is not differentiable at5.

We have

f(x) = |x5| =ifx5ifx<5.

The right-hand limit is

lim

x5+

f(x)f(5)

x5

=,

and the left-hand limit is

lim

x5

f(x)f(5)

x5

=.

Since these limits are not equal,f'(5) =lim

x5

f(x)f(5)

x5

does not exist andfis not differentiable at5.

Find a formula forf'

and sketch its graph.

f'(x) =ifx>5ifx<5

the blank spaces are boxes to fill in

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