
Applications
of differentiation  the graph of a function and its derivatives 
Points
of inflection 
Points
of inflection and concavity of the sine function 
Points
of inflection and concavity of the cubic polynomial 





Points
of inflection 
The
point of the graph of a function at which the graph crosses its tangent
and concavity changes from up to down
or vice versa is called the point of inflection. 
Therefore, at
the point of inflection the second derivative of the function is zero and changes its sign. 

Points
of inflection and concavity of the sine function 
For
example, the graph of f
(x)
= sin x
is concave
up inside the intervals 
(2k 
1)p
< x < 2kp,
k = 0,
+1,
+2,
. . .
since f ''
(x) = 
sin x > 0, 
it
is concave
down inside the intervals 
2kp
< x < (2k + 1)p,
k = 0,
+1,
+2,
. . .
since f ''
(x) = 
sin x < 0 
and
its points of inflection lie at x
= kp,
k = 0,
+1,
+2,
. . . , as shows the figure below. 

The first derivative
of a function at the point of inflection equals the slope of the tangent
at that point, so 
f ' (x) = cos
x
thus, m
=
f
' (kp)
= cos (kp)
= ±1,
k = 0,
+1,
+2,
. . . 

Point
of inflection and concavity of the cubic polynomial 
Let
examine concavity of graphs of cubic polynomial f (x) = a_{3}x^{3} + a_{2}x^{2}
+ a_{1}x + a_{0 }. 
The
coefficient a_{1}
of the source cubic defines three types of cubic functions. 
To obtain
the source form
we should plug the coordinates of translations 

into y
+ y_{0} = a_{3}(x
+ x_{0})^{3}
+ a_{2}(x + x_{0})^{2}
+ a_{1}(x + x_{0})
+ a_{0},
so we get 

To
find the abscissa of the point of inflection of the cubic polynomial we
should solve f ''
(x) = 0
for x, 
since f
' (x) = 3a_{3}x^{2} + 2a_{2}x
+ a_{1} 
then f ''
(x) = 6a_{3}x + 2a_{2},
so that f ''
(x) = 0
gives
x = 
a_{2}
/ (3a_{3})
= x_{0 }. 
Therefore,
the
abscissa of the point of inflection coincide
with the translation of the cubic in direction of the
xaxis.
Since
the first derivative
of a function at the point of inflection equals the slope of the tangent
at that point,
then 

Thus,
the value of tan a_{t}
= a_{1}
defines the three types of cubic functions as is shown in the figure
below. 

type
1 )
y

y_{0} = a_{3}(x

x_{0})^{3} 
and type
2 )
and
3 )
y 
y_{0} = a_{3}(x

x_{0})^{3}
+ a_{1}(x

x_{0}) 

1
)
tan a_{t}
= a_{1}
= 0 
2
)
tan a_{t}
= a_{1}
> 0 
3
)
tan a_{t}
= a_{1}
< 0 


Note
that all three graphs are drawn assuming a_{3}
> 0. 








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