Local and absolute extreme values ?Use the following graphs to identify the points on the?inter? val[?a,? t 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 domainD and let c be a fixed constant in D. Then the output value f ) 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 0ll less than f(x ). 0 That is , f(x) f(x ) 0 Thus, thegraph of f near x has a 0 eak at x 0 A function f has a local minimum or relative minimum at a point x if the valu0s f(x) of f for x ‘near’ x a re all greater than f(x ) . 0 0 That is f(x) f(x ). 0 Thus, the graph of f near x has 0 trough at x . (To mak0 the distinction clear, sometimes the ‘plain’ maximum and minimum are called absolute maximum and minimum). Step_3 The given graph is ; Now we need to verify the points , from the graph on the interval [a,b] at which the function has local and absolute extreme values . From the graph it is clear that the given function y = f(x) is continuous function on the given interval [a,b].Because the graph has no holes or break on that interval. So, the function is continuous on [a,b]. We know the result , that a function is continuous on the closed interval [a,b] has an absolute maximum value and an absolute minimum value on that interval . So , from the graph it is clear that the function y = f(x) is continuous in a closed interval [a,b]. so , local and absolute extreme values occurs in [a,b]. Hence , from the graph it is clear that p, q, r,and s are the interior points in [a,b], and a, b are the endpoints. Therefore , from the graph it is clearly mentioned absolute values occurs at end points ,and the local values occurs at the interior points. Hence, from the graph ; Local minimum value at x = q,s and Local maximum value at x= p,r. Absolute minimum value at x = a , and Absolute maximum value at x = b.