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

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

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,...\}`