Applications of differentiation - the graph of a function and its derivatives

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 < 2kpk = 0, +1, +2, . . .    since   f '' (x) = - sin x > 0,
it is concave down inside the intervals
2kp < x < (2k + 1)pk = 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) = a3x3 + a2x2 + a1x + a0 .
The coefficient a1 of the source cubic defines three types of cubic functions.
To obtain the source form we should plug the coordinates of translations
into        y + y0 = a3(x + x0)3 + a2(x + x0)2 + a1(x + x0) + a0,   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) = 3a3x2 + 2a2x + a1
then        f '' (x) = 6a3x + 2a2,   so that   f '' (x) = 0   gives   x = - a2 / (3a3) = x0 .
Therefore, the abscissa of the point of inflection coincide with the translation of the cubic in direction of the x-axis. 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 at = a1  defines the three types of cubic functions as is shown in the figure below.
 type  1 )    y - y0 = a3(x - x0)3 and  type  2 )  and  3 )    y - y0 = a3(x - x0)3 + a1(x - x0)
 1 )   tan at = a1 = 0 2 )   tan at = a1 > 0 3 )   tan at = a1 < 0
Note that all three graphs are drawn assuming  a3 > 0.
Functions contents F