flying along the path given by the curve f ( x ) = x + . A soldier, placed at the point ( , ), wants to shoot the helicopter when it is nearest to him. What is the nearest distance? In each of the above problem, there is something common, i.e., we wish to find out the maximum or minimum values of the given functions.
In order to tackle such problems, we first formally define maximum or minimum values of a function, points of local maxima and minima and test for determining such points. 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 f in I 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 I, 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 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 I and the point c is called an extreme point .
Remark In Fig . (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. Infact, through graphs, we can even find maximum/minimum value of a function at a