analyze and sketch a graph of the function over the given interval. label any intercepts, relative extrema…

Explanation:

Step1: Find x - intercepts

Set (y = f(x)=2x - 4\sin x=0). Since (2x - 4\sin x = 0), we can't solve it algebraically easily. Using a graphing - utility or numerical methods (e.g., Newton - Raphson method), in the interval (0\leq x\leq2\pi), when (x = 0), (y=2\times0 - 4\sin(0)=0). For other roots, we note that the function (y = 2x-4\sin x) is continuous. Another root in the interval ([0,2\pi]) is approximately (x\approx1.935). (y = 2\times1.935-4\sin(1.935)\approx0).

Step2: Find the first - derivative

Differentiate (y = f(x)=2x - 4\sin x) using the sum rule and the derivative formulas. (y^\prime=f^\prime(x)=2 - 4\cos x).

Step3: Find critical points

Set (y^\prime = 0), so (2 - 4\cos x=0). Then (\cos x=\frac{1}{2}). In the interval (0\leq x\leq2\pi), (x=\frac{\pi}{3}) and (x = \frac{5\pi}{3}).

Step4: Determine relative extrema

Find the second - derivative (y^{\prime\prime}=f^{\prime\prime}(x)=4\sin x). When (x=\frac{\pi}{3}), (y^{\prime\prime}( \frac{\pi}{3})=4\sin(\frac{\pi}{3}) = 2\sqrt{3}>0), so there is a relative minimum at (x=\frac{\pi}{3}). (y(\frac{\pi}{3})=2\times\frac{\pi}{3}-4\sin(\frac{\pi}{3})=\frac{2\pi}{3}-2\sqrt{3}\approx - 1.155). When (x=\frac{5\pi}{3}), (y^{\prime\prime}(\frac{5\pi}{3})=4\sin(\frac{5\pi}{3})=-2\sqrt{3}<0), so there is a relative maximum at (x = \frac{5\pi}{3}). (y(\frac{5\pi}{3})=2\times\frac{5\pi}{3}-4\sin(\frac{5\pi}{3})=\frac{10\pi}{3}+2\sqrt{3}\approx11.085).

Step5: Find points of inflection

Set (y^{\prime\prime}=4\sin x = 0). In the interval (0\leq x\leq2\pi), (x = 0,\pi,2\pi). But we usually consider non - endpoint points of inflection in an open interval. So the point of inflection in the open interval ((0,2\pi)) is (x=\pi), and (y(\pi)=2\pi-4\sin(\pi)=2\pi\approx6.283).

Step6: Find asymptotes

Since (f(x)=2x - 4\sin x) is a continuous function on the closed interval ([0,2\pi]), there are no vertical or horizontal asymptotes. So the equation of the asymptotes is DNE.

Answer:

Intercepts: ((x,y)=(0,0)) (smaller (x) - value), ((x,y)=(1.935,0)) (larger (x) - value) Relative minimum: ((x,y)=(\frac{\pi}{3},-1.155)) Relative maximum: ((x,y)=(\frac{5\pi}{3},11.085)) Point of inflection: ((x,y)=(\pi,6.283)) Asymptotes: DNE