Local Maximum
Local Minimum
No Extreme 

Suppose x_{0}, is a
critical point of a continuous function.
 If the value of the function's derivative changes from positive to
negative at x_{0}, then the function has a local maximum at
this point.
 If the value of the function's derivative changes from negative to
positive at x_{0}, then the function has a local minimum at
this point.
 If the value of the function's derivative does not change sign at
this point, then the function has no local extreme at this point.
The concept of the first derivative test can be better understand by
explaining the following example.
Examining the case when the derivative of a function changes from positive
to negative as shown on the left graphic. When x < c, the derivative
of the function is positive, according to the
Test
for Monotonic Functions the function is increasing. That is f(c) > f(x)
when x < c. When x > c, the derivative of the function is negative,
according to the
Test for Monotonic Functions the
function is decreasing. That is f(c) > f(x) when x > c. Thus, f(c)
is the local maximum according to the local
maximum definition.
No Extreme
