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 (x1) < f (x2) for each x1 < x2 in I .
A function  f is decreasing on an interval I if  f (x1) > f (x2) for each x1 < x2 in I .
Note that we always examine the graph of a function moving from left to right (x1 < x2) when determining the intervals in which the function increases or decreases.
Increasing/decreasing test
If the first derivative of a function  is positive for all x in an interval, then  f  is increasing on that interval.
If the first derivative of a function  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) = a2x2 + a1x + a0, 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) = 2a2x + a1  then   f ' (c) = 2a2c + a1,  and
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