Definition . Absolute maximum and absolute minimum A function f has an absolute maximum at c if f ( c ) ≥ f ( x ) for all x in domain of f . The number f ( c ) is called maximum value of f in the domain. Similarly f has an absolute minimum at c if f ( c ) ≤ f ( x ) for all x in domain of f and the number f ( c ) is called the minimum value of f on the domain.
The maximum and minimum value of f are called extreme values of f . NOTE Absolute maximum and absolute minimum values of a function f on an interval ( a , b ) are also called the global maximum and global minimum of f in ( a , b ). Criteria for local maxima and local minima Let f be a differentiable function on an open interval ( a , b ) containing c and suppose that f ll ( c ) exists. (i) If f l ( c ) = and f ll ( c ) > , then f has a local minimum at c .
(ii) If f l ( c ) = and f ll ( c ) < ,then f has a local maximum at c . NOTE In Economics, if y = f ( x ) represent cost function or revenue function, then the point at which dx dy = , the cost or revenue is maximum or minimum. Example . Find the extremum values of the function f ( x )= x + x – x .