Question

question 20 (5 points)
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which of the following are maximum and minimum points of the function y = 2 cos x - 1?
a) maximums: (0, 1), (2π, 1); minimum: (π, -3)
b) maximums: (0, 3), (2π, 3); minimum: (π, -1)
c) maximums: (0, 1) (π/2, 1); minimum: (3π/2, -3)
d) maximums: (0, 3), (2π, 3); minimum: (π, -3)

Explanation:

Step1: Recall range of cosine function

The range of the cosine function $y = \cos x$ is $[- 1,1]$.

Step2: Find range of $y = 2\cos x-1$

Multiply the range of $\cos x$ by 2: $2\cos x$ has a range of $[-2,2]$. Then subtract 1 from the range of $2\cos x$. So, $y=2\cos x - 1$ has a range of $[-2 - 1,2 - 1]=[-3,1]$.

Step3: Find maximum - points

The maximum value of $y = 2\cos x-1$ is 1. When $\cos x = 1$, $x = 2k\pi,k\in\mathbb{Z}$. For $k = 0,x = 0$ and $y=2\cos(0)-1=2\times1 - 1=1$; for $k = 1,x = 2\pi$ and $y=2\cos(2\pi)-1=2\times1 - 1=1$.

Step4: Find minimum - points

The minimum value of $y = 2\cos x-1$ is - 3. When $\cos x=-1$, $x=(2k + 1)\pi,k\in\mathbb{Z}$. For $k = 0,x=\pi$ and $y=2\cos(\pi)-1=2\times(-1)-1=-3$.

Answer:

A. Maximums: $(0,1),(2\pi,1)$; Minimum: $(\pi,-3)$