identify the inflection points and local maxima and minima of the graphed function. identify the open…

identify the inflection points and local maxima and minima of the graphed function. identify the open intervals on which the function is differentiable and is concave up and concave down\n\nfind the inflection point(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice\n\na. the point(s) is/are □\n(type an ordered pair. use a comma to separate answers as needed.)\nb. there are no inflection points

identify the inflection points and local maxima and minima of the graphed function. identify the open intervals on which the function is differentiable and is concave up and concave down\n\nfind the inflection point(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice\n\na. the point(s) is/are □\n(type an ordered pair. use a comma to separate answers as needed.)\nb. there are no inflection points

Answer

Explanation:

Step1: Find the second - derivative of the function

Given (y=-x + \sin(2x)), the first - derivative (y^\prime=-1 + 2\cos(2x)). The second - derivative (y^{\prime\prime}=-4\sin(2x)).

Step2: Solve (y^{\prime\prime}=0) for (x)

Set (y^{\prime\prime}=-4\sin(2x)=0). Then (\sin(2x) = 0), which gives (2x = k\pi), (x=\frac{k\pi}{2}), (k\in\mathbb{Z}). Since (-\frac{5\pi}{6}\leq x\leq\frac{5\pi}{6}), when (k = - 2), (x=-\pi) (not in the domain); when (k=-1), (x =-\frac{\pi}{2}) (not in the domain); when (k = 0), (x = 0); when (k = 1), (x=\frac{\pi}{2}) (not in the domain). Substitute (x = 0) into (y=-x+\sin(2x)), we get (y = 0).

Answer:

((0,0))