consider the equation below. (if an answer does not exist, enter dne.)\n\n$f(x)=3\\cos ^{2}(x)-6\\sin…

consider the equation below. (if an answer does not exist, enter dne.)\n\n$f(x)=3\\cos ^{2}(x)-6\\sin (x),\\quad 0\\leq x\\leq 2\\pi$\n\n(a) find the interval on which $f$ is increasing. (enter your answer using interval notation.)\n\nfind the interval on which $f$ is decreasing. (enter your answer using interval notation.)\n\n(b) find the local minimum and maximum values of $f$.\n\nlocal minimum value\n\nlocal maximum value\n\n(c) find the inflection points. (order your answers from smallest to largest $x$, then from smallest to largest $y$.)\n\n$(x,y)=$( )\n\n$(x,y)=$( )\n\nfind the interval on which $f$ is concave up. (enter your answer using interval notation.)\n\nfind the interval on which $f$ is concave down. (enter your answer using interval notation.)

consider the equation below. (if an answer does not exist, enter dne.)\n\n$f(x)=3\\cos ^{2}(x)-6\\sin (x),\\quad 0\\leq x\\leq 2\\pi$\n\n(a) find the interval on which $f$ is increasing. (enter your answer using interval notation.)\n\nfind the interval on which $f$ is decreasing. (enter your answer using interval notation.)\n\n(b) find the local minimum and maximum values of $f$.\n\nlocal minimum value\n\nlocal maximum value\n\n(c) find the inflection points. (order your answers from smallest to largest $x$, then from smallest to largest $y$.)\n\n$(x,y)=$( )\n\n$(x,y)=$( )\n\nfind the interval on which $f$ is concave up. (enter your answer using interval notation.)\n\nfind the interval on which $f$ is concave down. (enter your answer using interval notation.)

Answer

Explanation:

Step1: Find the first derivative

Use the chain - rule. If (y = 3\cos^{2}(x)-6\sin(x)), then (y^\prime=f^\prime(x)=3\times2\cos(x)(-\sin(x)) - 6\cos(x)=-6\cos(x)\sin(x)-6\cos(x)=-6\cos(x)(\sin(x) + 1))

Step2: Find critical points

Set (f^\prime(x)=0). Since (\sin(x)+1\geq0) for all (x), (\cos(x) = 0) gives (x=\frac{\pi}{2},\frac{3\pi}{2}) in the interval ([0,2\pi])

  • Test intervals:
    • For (x\in[0,\frac{\pi}{2})), let (x = 0), (f^\prime(0)=-6\times1\times(0 + 1)=-6<0)
    • For (x\in(\frac{\pi}{2},\frac{3\pi}{2})), let (x=\pi), (f^\prime(\pi)=-6\times(- 1)\times(0 + 1)=6>0)
    • For (x\in(\frac{3\pi}{2},2\pi]), let (x=\frac{7\pi}{4}), (f^\prime(\frac{7\pi}{4})=-6\times\frac{\sqrt{2}}{2}\times(\frac{-\sqrt{2}}{2}+1)=-6\times\frac{\sqrt{2}}{2}\times\frac{2 - \sqrt{2}}{2}<0)

Step3: Find the second derivative

(f^\prime(x)=-6\cos(x)\sin(x)-6\cos(x)=-3\sin(2x)-6\cos(x)) (f^{\prime\prime}(x)=-6\cos(2x)+6\sin(x)) Set (f^{\prime\prime}(x)=0), then (-6(1 - 2\sin^{2}(x))+6\sin(x)=0) (12\sin^{2}(x)+6\sin(x)-6 = 0), divide by (6): (2\sin^{2}(x)+\sin(x)-1=0) Let (t=\sin(x)), (2t^{2}+t - 1=(2t - 1)(t + 1)=0) (\sin(x)=\frac{1}{2}) or (\sin(x)=-1). In ([0,2\pi]), (x=\frac{\pi}{6},\frac{5\pi}{6},\frac{3\pi}{2})

  • Test intervals for concavity:
    • For (x\in[0,\frac{\pi}{6})), let (x = 0), (f^{\prime\prime}(0)=-6\times1+0=-6<0)
    • For (x\in(\frac{\pi}{6},\frac{5\pi}{6})), let (x=\frac{\pi}{2}), (f^{\prime\prime}(\frac{\pi}{2})=-6\times0 + 6\times1=6>0)
    • For (x\in(\frac{5\pi}{6},\frac{3\pi}{2})), let (x=\pi), (f^{\prime\prime}(\pi)=-6\times(-1)+0 = 6>0)
    • For (x\in(\frac{3\pi}{2},2\pi]), let (x=\frac{11\pi}{6}), (f^{\prime\prime}(\frac{11\pi}{6})=-6\times\frac{\sqrt{3}}{2}+6\times(-\frac{1}{2})=-3\sqrt{3}-3<0)

Step4: Evaluate function values

  • (f(\frac{\pi}{2})=3\cos^{2}(\frac{\pi}{2})-6\sin(\frac{\pi}{2})=-6)
  • (f(\frac{3\pi}{2})=3\cos^{2}(\frac{3\pi}{2})-6\sin(\frac{3\pi}{2})=6)
  • (f(\frac{\pi}{6})=3\cos^{2}(\frac{\pi}{6})-6\sin(\frac{\pi}{6})=3\times\frac{3}{4}-6\times\frac{1}{2}=-\frac{3}{4})
  • (f(\frac{5\pi}{6})=3\cos^{2}(\frac{5\pi}{6})-6\sin(\frac{5\pi}{6})=3\times\frac{3}{4}-6\times\frac{1}{2}=-\frac{3}{4})

Answer:

(a)

  • Increasing: ((\frac{\pi}{2},\frac{3\pi}{2}))
  • Decreasing: ([0,\frac{\pi}{2})\cup(\frac{3\pi}{2},2\pi]) (b)
  • Local minimum value: (-6)
  • Local maximum value: (6) (c)
  • Inflection points: ((\frac{\pi}{6},-\frac{3}{4}),(\frac{5\pi}{6},-\frac{3}{4}))
  • Concave up: ((\frac{\pi}{6},\frac{5\pi}{6}))
  • Concave down: ([0,\frac{\pi}{6})\cup(\frac{5\pi}{6},2\pi])