Question

For what values of theta (0
r=-3 + 4 cos(theta)? Note that the maximum r-value occurs at a point that is the maximum distance from the pole.

Answer 1

Answer:

$\theta = \pm \pi \cdot i$, $\forall i \in \{0,2,4,...\}$

Step-by-step explanation:

The determination of the maximum can be done with the help of the First and Second Derivative Tests. The first and second derivatives of the function are, respectively:

$r' = -4\cdot \sin \theta$

$r'' = -4\cdot \cos \theta$

First Derivative Test

$-4\cdot \sin \theta = 0$

$\sin \theta = 0$

Given the periodicity of the function, there are multiple solutions. The set of solutions is represented by:

$\theta = \pm \pi \cdot i$, $\forall i \in \mathbb{N}_{O}$

Second Derivative Test

There are multiple solutions. If $i$ is an odd number, then cosine is negative and output is positive, which means that associated value is an absolute minimum. Otherwise, if $i$ is zero or an even number, the cosine is positive and output is negative, which means that associated value is an absolute minimum. Consequently, the values of $\theta$ so that $r$ is a maximum are:

$\theta = \pm \pi \cdot i$, $\forall i \in \{0,2,4,...\}$