Sketch the graph of a function that has an absolute maximum, a local minimum, but no absolute minimum on [0, 3].

STEP_BY_STEP SOLUTION Step-1 When an output value of a function is a maximum or a minimum over the entire domain of the function, the value is called the absolute maximum or the absolute minimum. Let f be a function with domain D and let c be a fixed c onstant in D. Then the output value f ) is the 1. Absolute maximum value of f on D if and only if f(x) f(c) , for all x in D. 2. Absolute minimum value of f on D if and only if f(c) f(x) , for all x in D. Example ; The Absolute extreme values on a restricted domain ; If th domain of f(x ) = x is restricted to [-2, 3], the corresponding range is [0, 9]. As shown below, the graph on the interval [-2, 3] suggests that f has an absolute maximum of 9 at x = 3 and an absolu te minimum of 0 at x = 0. Step-2 be defined on the interval [a,b] , and x be the 0nterior point on [a,b]. A function f has a local maximum or relative maximum at a point x if the values 0 f(x) of f for x ‘near’ x are all less than f(x ). 0 0 That is , f(x) f(x )L0t f Thus, the graph of f near x has a 0 eak at x .0 A function f has a local minimum or relative minimum at a point x if the values 0 f(x) of f for x ‘near’ x a 0 re all greater than f(x 0 . That is f(x) f(x ). 0 Thus, the graph of f near x has a0trough at x . (To make0the distinction clear, sometimes the ‘plain’ maximum and minimum are called absolute maximum and minimum). Step-3 Now , we have to draw the function that has an absolute maximum, and local minimum, but no absolute minimum on [0, 3]. a). Example -1 In the above graph the function is continuous on (0,3) , because the graph has no holes or breaks then that interval the graph is continuous. In the above graph it is clear that the function has an absolute maximum at x=1 , and local minimum at x=2.5 . But no absolute minimum on [0,3]. Since the function is on that interval tends to infinity , there is no absolute minimum value in that interval. Hence , the above graph has an absolute maximum, and local minimum, but no absolute minimum on [0, 3]. b). Example-2 Graph of the given data is; In the above graph the function is continuous on (0,3) , because the graph has no holes or breaks then that interval the graph is continuous. In the above graph it is clear that the function has an absolute maximum at x=2 , and local minimum at x=0.5 . But no absolute minimum on [0,3]. Since the function is on that interval tends to infinity , there is no absolute minimum value in that interval. Hence , the above graph has an absolute maximum, and local minimum, but no absolute minimum on [0, 3].