

Applications
of differentiation  the graph of a function and its derivative 
Definition
of increasing and decreasing 
Increasing/decreasing
test 
Rolle's theorem






Definition
of increasing and decreasing of a function 
A
function f
is increasing on an interval I
if f (x_{1})
< f (x_{2})
for each x_{1}
< x_{2} in I
. 
A
function f
is decreasing on an interval I
if f (x_{1})
> f (x_{2})
for each x_{1}
< x_{2} in I
. 
Note
that we always examine the graph of a function moving from left to right (x_{1}
< x_{2})
when determining the intervals
in which the function increases or decreases. 

Increasing/decreasing
test 
If
the first derivative of a function f
is positive for all x
in an interval, then f
is increasing on that interval. 
If
the first derivative of a function f
is negative for all x
in an interval, then f
is decreasing on that interval. 


Rolle's
theorem 
If a
function f
is continuous on a closed interval [a,
b]
and differentiable between
a
and b,
for which the function
has the same value, that is f
(a)
= f (b),
then there exists point
c
inside the interval at which f
' (c) = 0,
as shows the left figure below. 


The
mean value theorem 
If a function f
is continuous on a closed interval [a,
b]
and differentiable between its endpoints, then
there is a point c
between a
and b
at which the slope of the tangent line to f
at c
equals the slope of the secant line through
the points (a,
f (a))
and (b,
f (b)), 

as
shows the right figure above. 
We
can also write the above formula as f
(b)

f (a) = f ' (c) (b 
a) 
and by substituting b
by x,
f
(x)
=
f (a) + f ' (c) (x 
a) 

Example:
Let
use the mean value theorem to prove that the abscissa of a point
of the parabola f
(x) = a_{2}x^{2} + a_{1}x
+ a_{0},
at
which the tangent is parallel to the secant line through
points (a,
f (a))
and (b,
f (b)),
is
the midpoint of the interval [a,
b].

Solution:
Since the derivative of the parabola
f
' (x) = 2a_{2}x + a_{1}
then f
' (c) = 2a_{2}c + a_{1},
and











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