Using the Definition to Compute the Derivative - Exercise 2
Exercise 2. Discuss the differentiability of
Answer. May be the scariest thing about this function is
the absolute value. So the best thing to do is to look for ways
to remove it. Therefore we are led to find out when x2 - x is
positive or negative. We get
Clearly the derivative exists at every point, except maybe at 0
and 1. Let us discuss these two points. Let us start with 0. We
Since the function is defined differently from the left and the
right of 0, then we have to consider the limits to the left and
to the right at 0. We have
This implies that f'(0) does not exist. Similar computations
will also give
which implies that f'(1) does not exist.
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