Local and absolute extreme values ?Use the following graphs to identify the points on the?inter? va? , b] ?at which local and absolute extreme values occur.

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 be0the 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 are all less than f(x ). 0 0 That is , f(x) f(x ) 0 Thus, the graph of f near x has a p eak 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 are all greater than f(x ) . 0 0 That is f(x) f(x 0. Thus, the graph of f near x ha0 a trough at x . (T0 make the distinction clear, sometimes the ‘plain’ maximum and minimum are called absolute maximum and minimum). Step_3 The given graph...