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

Concave up and concave down definition
Generalization of the mean value theorem, concavity of the graph of a curve
Let  f  be a function continuous on a closed interval [a, b] with f '(x) defined on that interval and  f ''(x) defined on (a, b), then
By substituting  a = x  and  b = x + h, where x is a given value and h is a variable quantity we get
To prove the formula we introduce new function
where Q is still undetermined.
Since j (b) = 0, we can define Q by setting j (a) = 0 too,
 what gives
Now, since j (x) at both endpoints of the interval is zero and has the derivative
j' (x) = - (b - x) f '' (x) + (b - x) Q
then, by Rolle's theorem, there exists a point c (a < c < b) such that  j' (c) = 0, that is,
- (b - c) f '' (c) + (b - c) Q = 0    or    Q = f '' (c),
and after equating both values of Q obtained is the generalized mean value formula to be proved.
Thus for example, to evaluate a function around the origin we should set x = 0 and substitute variable h by x
 into so we get
Concavity of the graph of a function
Concavity defines the shape or form of the graph of a function that is describes whether the graph is concave up (the cup opens upwards) or concave down (convex).
Concave up and concave down definition
A function is said to be concave up on an interval if its first derivative is increasing on the interval.
At the same time the second derivative of the function is positive on the interval.
A function is said to be concave down on an interval if its first derivative is decreasing on the interval.
At the same time the second derivative of the function is negative on the interval.
By following up changes of the slope of the tangent lines drawn to the graph of a function y = f (x), while moving along the positive direction of the x-axis, we get the information whether the derivative of  f is increasing or decreasing.
Let write the equation of the line tangent to the function at the point P (x0f (x0))
then the function values around x0
 where the term shows the distance of the curve from the tangent at the point x0 + h.
A function is said to be concave up on an interval if the graph of the function lies above the tangent in the neighborhood of P.
A function is said to be concave down on an interval if the graph of the function lies below the tangent in the neighborhood of P, as is shown in above figures.
Therefore, the graph of a function to be concave up at a point x = x0, the difference
f (x0 + h) -  yt (x0 + h)  must be positive, or   f '' (x0 ) > 0.
Similarly, the graph of a function is concave down at a point x = x0  if    f '' (x0 ) < 0.
Since the second derivative is derived from a given function by differentiating its first derivative then for example, the concavity condition f '' (x) > 0 shows that the graph of  f ' (x) increases while x variable passes through the point x0 moving in the positive direction of the x-axis.
At the same time, changes of the slope of the tangent line show whether the graph of a function increases or decreases passing through the positive or negative values.
That is, it depends on whether the angle between the tangent line and the positive direction of the x-axis is acute or obtuse and whether it increases or decreases while x increases.
In the figure below shown is how concavity of the graph of a function relates to the graphs of its first and the second derivative.
Calculus contents D