Using the Definition to Compute the Derivative  Exercise 3
Exercise 3. We say that the graph of f(x) has a cusp at (a,f(a)), if f(x) is continuous at a and if the
following two conditions hold:
 1.

as
from one side (left or right);
 2.

as
from the other side.
Determine whether
f(x) = x^{4/3} and
g(x) = x^{3/5} have a
cusp at (0,0).
Answer. For ,
we have
So
f(x) does not have a cusp at 0. In fact, the graph has a
horizontal tangent line at (0,0).
For the function g(x), we have
In this case, we have
So again (0,0) is not a cusp for g(x). But in this case, the
graph has a vertical tangent at this point. Remember that a
vertical line does not have a slope. So the derivative of g(x)at 0 does not exist.
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