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Consider the graph of the polar fimetion 
p=f(theta), where 
f(theta)=2=4cos theta, in the polar coordinate system for 
0 <= theta <= 2pi. Which of the following statements is true about the distance between the point with polar coordinates 
(f(theta),theta) and the origin?
(A) The distance is increasing for 
pi < theta < (5pi)/(3), because 
f(theta) is positive and increasing on the interval.
(B) The distance is increasing for 
(5pi)/(3) < theta < 2pi, because 
f(theta) is negative and increasing on the interval.
(C) The distance is decreasing for 
pi < theta < (5pi)/(3), because 
f(theta) is positive and decreasing on the interval,
(D) The distance is decreasing for 
(5pi)/(3) < theta < 2pi, because 
f(theta) is negative and deereasing on the interval.

Consider the graph of the polar fimetion p=f(θ) p=f(\theta) , where f(θ)=2=4cosθ f(\theta)=2=4 \cos \theta , in the polar coordinate system for 0θ2π 0 \leq \theta \leq 2 \pi . Which of the following statements is true about the distance between the point with polar coordinates (f(θ),θ) (f(\theta), \theta) and the origin?\newline(A) The distance is increasing for π<θ<5π3 \pi<\theta<\frac{5 \pi}{3} , because f(θ) f(\theta) is positive and increasing on the interval.\newline(B) The distance is increasing for 5π3<θ<2π \frac{5 \pi}{3}<\theta<2 \pi , because f(θ) f(\theta) is negative and increasing on the interval.\newline(C) The distance is decreasing for π<θ<5π3 \pi<\theta<\frac{5 \pi}{3} , because f(θ) f(\theta) is positive and decreasing on the interval,\newline(D) The distance is decreasing for 5π3<θ<2π \frac{5 \pi}{3}<\theta<2 \pi , because f(θ) f(\theta) is negative and deereasing on the interval.

Full solution

Q. Consider the graph of the polar fimetion p=f(θ) p=f(\theta) , where f(θ)=2=4cosθ f(\theta)=2=4 \cos \theta , in the polar coordinate system for 0θ2π 0 \leq \theta \leq 2 \pi . Which of the following statements is true about the distance between the point with polar coordinates (f(θ),θ) (f(\theta), \theta) and the origin?\newline(A) The distance is increasing for π<θ<5π3 \pi<\theta<\frac{5 \pi}{3} , because f(θ) f(\theta) is positive and increasing on the interval.\newline(B) The distance is increasing for 5π3<θ<2π \frac{5 \pi}{3}<\theta<2 \pi , because f(θ) f(\theta) is negative and increasing on the interval.\newline(C) The distance is decreasing for π<θ<5π3 \pi<\theta<\frac{5 \pi}{3} , because f(θ) f(\theta) is positive and decreasing on the interval,\newline(D) The distance is decreasing for 5π3<θ<2π \frac{5 \pi}{3}<\theta<2 \pi , because f(θ) f(\theta) is negative and deereasing on the interval.
  1. Analyze Function Behavior: Analyze the function f(θ)=24cos(θ)f(\theta) = 2 - 4\cos(\theta) to understand its behavior over the interval 0θ2π0 \leq \theta \leq 2\pi.
  2. Determine Increasing/Decreasing Intervals: Determine the intervals where f(θ)f(\theta) is increasing or decreasing.
  3. Match Behavior with Statements: Match the behavior of f(θ)f(\theta) with the given statements.

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