11-14. Absolute maximum/minimum values from graphs Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum value or an absolute minimum value.
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 The given graph is ; Now we need to verify the points , from the graph on the interval [a,b] at which the function has an absolute maximum value or an absolute minimum value. From the graph it is clear that the given function y = f(x) is not continuous function on the given interval [a,b].Because the graph has a break at x=c , here c is the interior point of [a,b]. So, the function is not continuous on [a,b]. Observe the graph the given interval is not a closed interval , at x=c the function has a break. So , from the graph it is clear that the function y = f(x) is not in closed interval. so , no absolute maximum value and no absolute minimum value (here the maximum and minimum values are not mentioned).Because the existence of extreme values depend on the both the function and the interval , so the function is not in the closed interval then the function may not attains the absolute extreme values.