Sketch the graph of a fun?ction ?f that has a local minimum value at a point w? ? er? '? )is undefined.

STEP_BY_STEP SOLUTION Step-1 be defined on the interval [a,b] , and x be0the interior 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 a0e all less than f(x ). 0 That is , f(x) f(x 0Let f Thus, the graph of f nea r x has a peak at x . 0 0 A function f has a local minimum or relative minimum at a point x if th0 values f(x) of f for x ‘near’ x a0 re all greater than f(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_2 Let f be a continuous function defined on an open interval containing a number ‘c’.The number ‘c’ is critical value ( or critical number ). If f (c) = 0 1 or f (c) is undefined. A critical point on that graph of f has the form (c,f(c)). Step-3 Example-1 Graph of the given data is ; The given function is y = f(x) =|x| = { x , if x 0 = { -x , if x<0. 1 Here f (x) is does not exist , since left hand derivative is -1 , and right hand derivative is 1. But , the function attains the local minimum at x=0. Example-2 In the given graph it is clear that the function is continuous . The function has local minimum at x= c , and at ‘c’ , f (x) does not exist.