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 = 7 x + 7 sin x, 0 leq x leq 2 pi )\n\nidentify the coordinates of the local maximum points. select the correct choice below and,\nif necessary, fill in the answer box to complete your choice.\n\na. the local maximum point(s) is/are ( ( 2 pi, 14 pi ) )\n(type an ordered pair. type an exact answer in terms of ( pi ). use a comma to separate\nanswers as needed. do not use commas in the individual coordinates.)\nb. there are no local maximum points.\n\nidentify the coordinates of the local minimum points. select the correct choice below and,\nif necessary, fill in the answer box to complete your choice.\n\na. the local minimum point(s) is/are\n(type an ordered pair. type an exact answer in terms of ( pi ). use a comma to separate\nanswers as needed. do not use commas in the individual coordinates.)\nb. there are no local minimum points

identify the coordinates of any local and absolute extreme points and inflection points. graph the function.\n\n( y = 7 x + 7 sin x, 0 leq x leq 2 pi )\n\nidentify the coordinates of the local maximum points. select the correct choice below and,\nif necessary, fill in the answer box to complete your choice.\n\na. the local maximum point(s) is/are ( ( 2 pi, 14 pi ) )\n(type an ordered pair. type an exact answer in terms of ( pi ). use a comma to separate\nanswers as needed. do not use commas in the individual coordinates.)\nb. there are no local maximum points.\n\nidentify the coordinates of the local minimum points. select the correct choice below and,\nif necessary, fill in the answer box to complete your choice.\n\na. the local minimum point(s) is/are\n(type an ordered pair. type an exact answer in terms of ( pi ). use a comma to separate\nanswers as needed. do not use commas in the individual coordinates.)\nb. there are no local minimum points

Answer

Explanation:

Step1: Find the first derivative

Differentiate (y = 7x+7\sin x) using the sum rule ((u + v)^\prime=u^\prime + v^\prime) where (u = 7x) and (v = 7\sin x). The derivative of (u = 7x) is (u^\prime=7), and the derivative of (v = 7\sin x) is (v^\prime = 7\cos x). So (y^\prime=7 + 7\cos x).

Step2: Find the critical points

Set (y^\prime = 0), so (7+7\cos x=0). Divide both sides by (7): (1+\cos x = 0), then (\cos x=- 1). For (0\leq x\leq2\pi), (x = \pi).

Step3: Find the second derivative

Differentiate (y^\prime=7 + 7\cos x). The derivative of (7) is (0) and the derivative of (7\cos x) is (-7\sin x). So (y^{\prime\prime}=-7\sin x).

Step4: Classify the critical point

Evaluate (y^{\prime\prime}) at (x = \pi): (y^{\prime\prime}(\pi)=-7\sin(\pi)=0). Now, check the sign of (y^\prime) around (x=\pi). For (x\in[0,\pi)), let (x=\frac{\pi}{2}), (y^\prime(\frac{\pi}{2})=7 + 7\cos(\frac{\pi}{2})=7>0). For (x\in(\pi,2\pi]), let (x=\frac{3\pi}{2}), (y^\prime(\frac{3\pi}{2})=7+7\cos(\frac{3\pi}{2})=7>0). Since (y^\prime) does not change sign at (x = \pi), there is no local minimum.

Answer:

B. There are no local minimum points.