12. which of the following is the graph of the polar function r = f(θ), where f(θ) = 2cos(3θ), in the polar…

Explanation:

Step1: Analyze key - points of polar function

The general form of a polar function is (r = a\cos(n\theta)). For (r = 2\cos(3\theta)), when (n = 3) (an odd number), the number of petals is (n=3).

Step2: Evaluate function at boundary values

When (\theta=\frac{3\pi}{2}), (r = 2\cos(3\times\frac{3\pi}{2})=2\cos(\frac{9\pi}{2}) = 0). When (\theta=\frac{11\pi}{6}), (r = 2\cos(3\times\frac{11\pi}{6})=2\cos(\frac{11\pi}{2}) = 0). Also, we can find some intermediate - values. For example, when (\theta=\frac{5\pi}{3}), (r = 2\cos(3\times\frac{5\pi}{3})=2\cos(5\pi)= - 2).

Step3: Match with graphs

By analyzing the behavior of the function (r = 2\cos(3\theta)) in the interval (\frac{3\pi}{2}\leq\theta\leq\frac{11\pi}{6}), we can match it with the correct graph.

Answer:

(Please provide the correct option based on the above - mentioned analysis. Since the option texts are not clearly visible in the image, assume the correct option is determined through the above steps. For example, if after analysis, option C is correct, the answer would be) C.