using the given graph of the function f, find the following.\n(a) the numbers, if any, at which f has a…
using the given graph of the function f, find the following.\n(a) the numbers, if any, at which f has a local\nmaximum. what are these local maximum values?\n(b) the numbers, if any, at which f has a local\nminimum. what are these local minimum values?\n(a) find the number(s) x at which f has a local maximum. select the correct choice and, if necessary, fill in the answer\nbox to complete your choice.\noa. x=\n(type an exact answer, using π as needed. use a comma to separate answers as needed.)\nob. there is no local maximum.\nfind the local maximum value. select the correct choice and, if necessary, fill in the answer box to complete your choice.\noa. the local maximum value is\n(type an exact answer, using π as needed. use a comma to separate answers as needed.)\nob. there is no local maximum.\n(b) find the number(s) x at which f has a local minimum. select the correct choice and, if necessary, fill in the answer
Answer
Explanation:
Step1: Recall the definition of local maximum
A local maximum of a function (y = f(x)) occurs at a point (x = a) if (f(a)\geq f(x)) for all (x) in some open interval containing (a). Looking at the graph, the function (y = f(x)) has a peak at (x=\frac{\pi}{2}).
Step2: Determine the local maximum value
The (y -)coordinate of the peak (local maximum) is (y = 8).
Step3: Recall the definition of local minimum
A local minimum of a function (y = f(x)) occurs at a point (x = b) if (f(b)\leq f(x)) for all (x) in some open interval containing (b). Looking at the graph, the function (y = f(x)) has a trough at (x=-\frac{\pi}{2}).
Answer:
(a)
- For the number (x) at which (f) has a local maximum: (A. \ x=\frac{\pi}{2})
- For the local maximum value: (A. \ The\ local\ maximum\ value\ is\ 8)
(b)
- For the number (x) at which (f) has a local minimum: (A. \ x =-\frac{\pi}{2})
- For the local minimum value: (A. \ The\ local\ minimum\ value\ is - 8)