Definition : Let f be a function defined on an interval I. Then
(a) f is said to have a maximum value in I, if there exists a point c in I such that f(c) > f(x), for all x ∈ I
The number f(c) is called the maximum value of/in 1 and the point c is called a point of maximum value of f in I.
(b) f is said to have a minimum value in l, if there exists a point c in I such that f(c) < f(x), for all x ∈ I.
The number f(c), in this case, is called the minimum value of f in I and the point c, in this case, is called a -point of minimum x value of f in I.
(c) f is said to have an extreme value in I if there exists a point c in I such that f(c) is either a maximum value or a minimum value of f in I.
The number f(c), in this case, is called an extreme value of f in 1 and the point c is called an extreme point.
Remark : In figure (a), (b) and (c), we have exhibited that graphs of certain particular functions help us to find maximum value and minimum value at a point. In fact, through graphs, we can even find maximum f minimum value of a function at a point at which it is not even differentiable.
Example: Find the maximum and the minimum value, if any, of the function/given by
f(x) = x2, x ∈ R
Sol. From the graph of the given function, we have f(x) = 0 if x = 0. Also
f(x) ≥ 0, for all x ∈ R
Therefore, the minimum value of f is 0 and the point of minimum value of f is x = 0. Further, it may be observed from the graph ot the function that f has no maximum value and hence no point of maximum value of f in R,
Note : If we restrict the domain of f to [- 2, 1], only then/will have maximum value (- 2)2 = 4 at x = - 2.
Monotonic function: The function which is maximum or minimum at the end points of the defined domain is called monotonic function.
Monotonic increasing function : The function which is increasing in its defined domain, is called monotonic increasing function.
Monotonic decreasing function : The function which is deceasing in its defined domain is called monotonic decreasing function.
Note : By a monotonic function f in an interval I, we mean that f is either increasing in I or decreasing in I.
