To determine the values of [tex]\(\theta\)[/tex] where the maximum [tex]\(r\)[/tex]-values occur for the given polar equation [tex]\(r = -3 + 4 \cos \theta\)[/tex], we will analyze the equation step-by-step and consider the behavior of [tex]\(\cos \theta\)[/tex].
1. Consider the Range of [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos \theta \text{ varies between -1 and 1.}
\][/tex]
2. Find the expression for [tex]\(r\)[/tex]:
[tex]\[
r = -3 + 4 \cos \theta.
\][/tex]
3. Analyze the Conditions for Maximum [tex]\(r\)[/tex]:
[tex]\[
\text{To maximize } r, \text{ we need to maximize } 4 \cos \theta.
\][/tex]
[tex]\[
\cos \theta \text{ achieves its maximum value of 1 when } \theta \text{ is } 0\text{ or } 2\pi.
\][/tex]
4. Calculate [tex]\(r\)[/tex] at [tex]\(\theta = 0\)[/tex]:
[tex]\[
r = -3 + 4 \cos(0) = -3 + 4 \cdot 1 = 1.
\][/tex]
[tex]\[
\theta = 0 \text{ (or } 2\pi \text{ as } \cos(2\pi) = 1 \text{ again)} \text{ gives the maximum value of \(r\).}
\][/tex]
So, the maximum [tex]\(r\)[/tex]-value occurs at [tex]\(\theta = 0\)[/tex].
Given the choices:
a. [tex]\(0, \pi\)[/tex]
b. [tex]\(0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}\)[/tex]
c. [tex]\(0\)[/tex]
d. [tex]\(\pi\)[/tex]
The correct answer is [tex]\(\boxed{C}\)[/tex].
