identify the coordinates of any local and absolute extreme points and inflection points. graph the…

identify the coordinates of any local and absolute extreme points and inflection points. graph the function.\n\n$y = 8x + 8\\sin x$, $0 \\leq x \\leq 2\\pi$\n\nidentify the coordinates of the local maximum points. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\n○ a. the local maximum point(s) is/are\n(type an ordered pair. type an exact answer in terms of $\\pi$. use a comma to separate answers as needed. do not use commas in the individual coordinates.)\n\n○ b. there are no local maximum points.

identify the coordinates of any local and absolute extreme points and inflection points. graph the function.\n\n$y = 8x + 8\\sin x$, $0 \\leq x \\leq 2\\pi$\n\nidentify the coordinates of the local maximum points. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\n○ a. the local maximum point(s) is/are\n(type an ordered pair. type an exact answer in terms of $\\pi$. use a comma to separate answers as needed. do not use commas in the individual coordinates.)\n\n○ b. there are no local maximum points.

Answer

Explanation:

Step1: Find the first derivative

The derivative of (y = 8x+8\sin x) is (y'=8 + 8\cos x). Set (y'=0), so (8+8\cos x = 0), which simplifies to (\cos x=- 1).

Step2: Solve for (x) in the given interval

In the interval (0\leq x\leq2\pi), when (\cos x=-1), (x = \pi). Now find the second derivative (y''=-8\sin x). When (x = \pi), (y''=-8\sin\pi=0). We can also check the sign of (y') around critical points. Let's take a test - point (x=\frac{\pi}{2}) (to the left of (x = \pi) in the interval ([0,2\pi])), (y'=8 + 8\cos\frac{\pi}{2}=8>0). Take a test - point (x=\frac{3\pi}{2}) (to the right of (x = \pi) in the interval ([0,2\pi])), (y'=8+8\cos\frac{3\pi}{2}=8>0).

Since the function is increasing on both sides of (x = \pi) (because (y'>0) for (x\in[0,\pi)) and (x\in(\pi,2\pi])), there is no local maximum.

Answer:

B. There are no local maximum points.