using the given graph of the function f, find the following. (a) the numbers, if any, at which f has a local…
using the given graph of the function f, find the following. (a) the numbers, if any, at which f has a local maximum. what are these local maximum values? (b) the numbers, if any, at which f has a local minimum. what are these local minimum values? (a) find the number(s) x at which f has a local maximum. select the correct choice and, if necessary, fill in the answer a. x= (type an exact answer, using π as needed. use a comma to separate answers as needed.) b. there is no local maximum.
Answer
Explanation:
Step1: Recall local - maximum definition
A function (y = f(x)) has a local maximum at (x = c) if (f(c)\geq f(x)) for all (x) in some open interval containing (c).
Step2: Analyze the graph
From the given graph of the function (y = f(x)), we can see that the function has a peak at the point ((\frac{\pi}{2},9)). In an open - interval around (x=\frac{\pi}{2}), the value of (y = f(x)) is (9) and for other (x) values in that open interval, (f(x)<9).
Step3: Recall local - minimum definition
A function (y = f(x)) has a local minimum at (x = c) if (f(c)\leq f(x)) for all (x) in some open interval containing (c).
Step4: Analyze the graph for local minimum
From the graph, the function has a trough at the point ((-\frac{\pi}{2}, - 9)). In an open - interval around (x =-\frac{\pi}{2}), the value of (y = f(x)) is (-9) and for other (x) values in that open interval, (f(x)>-9).
Answer:
(a) (x=\frac{\pi}{2}), local maximum value is (9) (b) (x =-\frac{\pi}{2}), local minimum value is (-9)