Function in a Closed Interval Let us consider a function f given by f ( x ) = x + , x ∈ ( , ) Observe that the function is continuous on ( , ) and neither has a maximum value nor has a minimum value. Further, we may note that the function even has neither a local maximum value nor a local minimum value. However, if we extend the domain of f to the closed interval [ , ], then f still may not have a local maximum (minimum) values but it certainly does have maximum value = f ( ) and minimum value = f ( ). The maximum value of f at x = is called absolute maximum value ( global maximum or greatest value ) of f on the interval [ , ].
Similarly, the minimum value of f at x = is called the absolute minimum value ( global minimum or least value ) of f on [ , ]. Consider the graph given in Fig . of a continuous function defined on a closed interval [ a , d ]. Observe that the function f has a local minima at x = b and local Fig .
minimum value is f ( b ). The function also has a local maxima at x = c and local maximum value is f ( c ). Also from the graph, it is evident that f has absolute maximum value f ( a ) and absolute minimum value f ( d ). Further note that the absolute maximum (minimum) value of f is different from local maximum (minimum) value of f .
We will now state two results (without proof) regarding absolute maximum and absolute minimum values of a function on a closed interval I. Theorem Let f be a continuous function on an interval I = [ a , b ]. Then f has the absolute maximum value and f attains it at least once in I. Also, f has the absolute minimum value and attains it at least once in I.