identify the points for 0 ≤ θ < 2π where the maximum r - value occurs on the graph of the polar equation. r…

Explanation:

Step1: Analyze the cosine function

The range of $\cos\theta$ is $[- 1,1]$. We want to maximize $r = 3+2\cos\theta$.

Step2: Find the maximum - value of $\cos\theta$

Since $r = 3+2\cos\theta$, to maximize $r$, we need to maximize $\cos\theta$. The maximum value of $\cos\theta$ is $1$.

Step3: Solve for $\theta$

Set $\cos\theta = 1$. We know that $\cos\theta=1$ when $\theta = 0$ for $0\leq\theta<2\pi$. When $\theta = 0$, $r=3 + 2\times1=5$.

Answer:

The maximum $r$-value occurs at the point $(\theta = 0,r = 5)$ in polar coordinates. In the form of just the angle for the given domain, $\theta = 0$.