Sketch the graph of a fun?ction ? that has a local maximum value at a point ?c w?? ere ?? )= 0.

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 1 number ‘c’.The number ‘c’ is critical value ( or critical number ). If f (c) = 0 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 ; In the above graph the given function is polynomial function , and it is continuous for all values of x. , because the graph has no holes or breaks then that interval the graph is continuous. 3 f(x) = x -3x +6. f (x) = 3x 3 1 2 f (x) = 0 = 3x 3 3(x-1)(x+1) = 0 x= -1 ,and x = 1. 3 At x= -1 , then f(-1) = (1) -3(-1) +6 = -1 +3+6 = 8 . At x= 1 , then f(1) = (1) -3(1) +6 = 1 -3+6 = 4 . Hence, the critical points of f (x) are (1 ,8) , and (1,4). At , x= -1 the function has local maximum. Example -2 In the above graph the given function is continuous , because the graph has no holes 1 or breaks. Here the function has local maximum at c=0 , and f (c) = 0.