The First Derivative and the Shape of a Function f(x)
Read this section to learn how the first derivative is used to determine the shape of functions. Work through practice problems 1-9.
Using the Derivative to Test for Extremes
The first derivative of a function tells about the general shape of the function, and we can use that shape information to determine if an extreme point is a maximum or minimum or neither.
First Derivative Test for Local Extremes
Let be a continuous function with
or
is undefined.
(i) If (left of
)
and
(right of
)
, then
is a local maximum (Fig. 19a)
(ii) If (left of
)
and
(right of
)
, then
is a local minimum (Fig. 19b)
(iii) If (left of
)
and
(right of
)
, then
is not a local extreme (Fig. 19c)
(iv) If (left of
)
and
(right of
)
, then
is not a local extreme (Fig. 19d)
Fig. 19
Practice 9: Find all extremes of and use the First Derivative Test to determine if they are maximums, minimums or neither.
A variant of the First Derivative Test can also be used to determine whether an endpoint gives a maximum or minimum for a function.