Designing a function ?Sketch the graph of a continuous functio ? n f on [0, 4] satisfying the given properties. ? ? f'(??x)=0 for ?x =?1 and 2? has an absolute maxi?mum?at ?x = 4; f ? has an absolute min ? imum at ? ? = 0; and? has a local min? imum at ?x =2.
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 constant in D. Then the output value f(c) 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 0he interior point on [a,b]. A function f has a local maximum or relative maximum at a point x 0 if the values f(x) of f for x ‘near’ x a0e all less than f(x ). 0 That is , f(x) f(x )0 Thus, the graph of f near x has a eak at x . 0 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 a0e all greater than f(x )0. That is f(x) f(x 0. Thus, the graph of f near x has a trough at x . (To make the distinction clear, 0 0 sometimes the ‘plain’ maximum and minimum are called absolute maximum and minimum). Step_3 Given is ; ‘f’ be a continuous function on [0,4] , and given properties are ; f (x) = 0 , for x=1 and 2 , f has an absolute maximum at x=4 , an absolute minimum at x=0, and f has a local minimum at x=2. 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 are the interior points on the interval. 1 Given ; f (x) = 0 , for x= 1 and 2 .so the graph of f(x) has slope is zero at x=1, 2 , and also given that ; f has an absolute maximum at x=4 , so f(4) f(x) for all x in [0,4].That is f(4) is an absolute maximum value at x=4 , since from the definition step-1. f has an absolute minimum at x=0 , so f(x) f(0) for all x in [0,4].That is f(0) is an absolute minimum value at x=0, since from the definition step-1. f has local minimum at x=2 , so f(x) f(2) for all x in 2 belongs to [0,4].That is f(2) is local minimum value at x=2, since from the definition step-2. Step-5 Therefore , the related graph of the function y = f(x) is ;