Proofs of Applications
Theorem:
If 
is continuous on
an interval I and if 
for each 
in the interior
of I, then 
is increasing on
I.
Similarly, if 
is continuous on
I and
for all interior points x in I, then f(x)
is decreasing on I.
Proof:
To prove the first part, we apply the theorem of the
Mean value to two points
in I, we find that 
, where 
is between 
in I.
For all 
when 
is positive, we
obtain 
, and so 
. Then f(x) is increasing.
The proof of the second part of the theorem is
analogous.
Now to find the increase and the decrease intervals
of the function, we have to study the sign of its first derivative. In many
cases, that may be difficult. The next theorem helps us to study the sign of
the derivative at a whole interval by finding its sign at only one point.
Theorem of the first derivative test:
1-
If f
is increasing 
in some interval
to the left of 
where 
is an endpoint
of this interval, and if f is decreasing 
in some interval
to the right of 
where 
is an endpoint, then f has a relative (local) maximum at 
if the function
is continuous there.
2-
If f
is decreasing 
in some interval
to the left of 
where 
is an endpoint
of this interval, and if f is increasing 
in some interval
to the right of 
where 
is an endpoint, then f has a relative (local) minimum at 
if the function
is continuous there.
Proof:
Let 
interval to the left of 
,
=interval to the right of 
, and 
.
If f is increasing on 
and decreasing on
and both contain the point 
, we must have 
for all x in I.
The proof of (2) is the same.