Monotonicity, Maxima and Minima
6.0 Maxima and Minima
6.0 Maxima and Minima
While making graphs of functions, we have seen a highest point or the lowest point in it. In this topic, we will study the 
concept of derivative to calculate the value at which we get the highest point (maxima) and the lowest point (minima) of various functions.
concept of derivative to calculate the value at which we get the highest point (maxima) and the lowest point (minima) of various functions.For better understanding, let us assume a function $f$ in the open interval $I \in (a,b)$. From the figure, we can say that
- $f$ is having the maximum value at point $C$ in the interval $I \in (a,b)$ which is only possible when $$f(c) \geqslant f(x)$$ The point on the $X$-axis corresponding to $C$ i.e., $'c'$ is the point of maximum value of $f$ in the interval $I$.
- $f$ is having the minimum value at point $D$ in the interval $I$ which is only possible when $$f(d) \leqslant f(x)$$ The point on the $X$-axis corresponding to $D$ i.e., $'d'$ is the point of minimum value of $f$ in the interval $I$.
- $f$ is having an extreme value if there exists a point in $I$ such that $f(x)$ is either a maximum value or a minimum value of $f$ in $I$. This point is called the extreme point.
Note:
This concept of finding maximum and minimum value of a function at a particular point is applicable even to those functions which are not differentiable at that point.
But remember that it is the graphical approach and only applicable when in the question it is asked to find the maximum and minimum value of the function in the given open interval (means the end values of the interval are not the part of domain of the function).
