Designing a function S ? ketch the graph of a continuous function ? f on [0, 4] ?satisfying the given properties. ? ? f'(?x? )= 0 at ?x = 1?and 3? '(2)is und?efined; ?f has an absolute maxim ? um at x? =2; ? f has neither a local maximum nor a local mini?mum al ?x? =1; and ?f has an absolute min ? imum at ?x =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 domainD and let c be a fixed constant 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. Step-2 Let f 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 are0all less than f(x ). 0 That is , f(x) f(x ) 0 Thus, the graph of f near x h0s a p eak at x .0 A function f has a local minimum or relative minimum at a point x if the value0 f(x) of f for x ‘near’ x a re all greater than f(x ) . 0 0 That is f(x) f(x ). 0 Thus, the graph of f near x h0s a trough at x . (0o make the distinction clear, sometimes the ‘plain’ maximum and minimum are called absolute maximum and minimum). Step_3 Given is ; 1 ‘f’ be a continuous function on [0,4] , and given properties are ; f (x) = 0 1 at x= 1 ,3 and f (2) is undefined, f has an absolute maximum at x=2 , an absolute minimum at x=3, and f has neither local maximum nor local minimum at x=1 Now , we need to sketch the graph of f(x) on [0,4]. Step-4 Let , ‘f’ be a continuous function on [0,4] . so, the graph of f(x) is continuous on [0,4].From the given it is clear that 0, 4 are the endpoints of the interval , and 1,2 ,3are the interior points on the interval. Given ; f (x) = 0, at x= 1 ,3 . So the graph of f has a slope zero at 1 x=1,3. f (2) is undefined that is the graph of f(x) is vertical line at x=2 ( vertical line means parallel to y-axis),and also given that ; f has an absolute maximum at x=2 , so f(2) f(x) for all x in [0,4].That is f(2) is an absolute maximum value at x= 2, since from the definition step-1. f has an absolute minimum at x=3 , so f(x) f(3) for all x in [0,4].That is f(3) is an absolute minimum value at x=3, since from the definition step-1. F has a neither local maximum nor local minimum at x=1 Step-5 Therefore , the related graph of the function y = f(x) is ;
