Designing a function S ? ketch the graph of a continuous functio ? n f on [0, 4] ?satisfying the given properties. ? f? is unde?fined at ?x = ? and 3;? ?f?(2)=0;? has a local maxi?mum?at x ? = 1; f? has a local min ? imum? at ?x = 2; ?f has an absolute max ? imum al ?x? = 3; and ?f has an absolute mi ? nimum at ?x =4.

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 o utput value ) 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 are all less than f(x ). 0 0 That is , f(x) f(x ) 0 Thus, the grap h of f near x ha0 a p eak at x 0 A function f has a local minimum or relative minimum at a point x if the values0 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 h0s a trough at x . (T0 make the distinction clear, 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) is 1 undefined at x= 1 ,3 and f (2) = 0, f has an absolute maximum at x=3 , an absolute minimum at x=4, and f has local maximum at x=1 , 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 ,3are the interior points on the interval. Given ; f (x) is undefined at x= 1 ,3 . So the slope of f(x) is undefined at x=1, 3. That is the graph of f(x) is vertical line at x=1,3 ( vertical line means parallel 1 to y-axis)and f (2) = 0 .so the graph of f(x) has slope is zero at x= 2 , and also given that ; f has an absolute maximum at x=3 , so f(3) f(x) for all x in [0,4].That is f(3) is an absolute maximum value at x= 3, since from the definition step-1. f has an absolute minimum at x=4 , so f(x) f(4) for all x in [0,4].That is f(1) is an absolute minimum value at x=1, since from the definition step-1. f has local minimum value at x=2 .So , 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. f has local maximum value at x=1 .So , so f(1) f(x) for all x in 1 belongs to [0,4].That is f(1) is local maximum value at x=1, since from the definition step-2. Step-5 Therefore , the related graph of the function y = f(x) is ;