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I'm answering the following question:

Find the maximum and minimum values for the function f(x) = 2 / (1 + 3cos(θ) + 4sin(θ))

I have found that the answer shows that the minimum occurs at the point (53.1 degrees, 1/3) and the maximum occurs at (233.1 degrees, -0.5).

This doesn't really intuitively make much sense to me. How is it possible that the minimum occurs above the maximum on the graph? I understand that the aforementioned function has no absolute maximum or minimum, but doesn't it make sense that locally the minimum would be lower than the maximum point?

I am working through a book and that is the answer given.

To get to such, 3cos(θ) + 4sin(θ) is put into harmonic form, giving 5sin(t+36.9).

From 2 / (1 + 5sin(θ+36.9)), the knowledge that f(x) will obtain a maximum when sin(θ+36.9) = 1 and a minimum when sin(t+36.9) = -1 is used to get to the answer by solving for (θ+36.9) = arcsin(1) and (θ+36.9)=arcsin(-1), respectively. However, it puts the local maximum below the local minimum.

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    None of your analysis makes any sense, I agree. Local minima can be greater than local maxima, but if the graph has none... can you post your calculations in the body of your question? – 2017-01-23
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    is there a interval given? – 2017-01-23
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    I have added extra detail into the body, sorry for the initial lack of information. – 2017-01-23

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That can easily happen because of discontinuities of the function. The simplest example I know is the graph of $f(x)=x+\frac{1}{x}$. It consists of two separate branches, and the one with a minimum happens to lie higher than the one with a maximum:

the graph of $f(x)=x+\frac{1}{x}$

The same is the case with your function:

the graph of $f(x)=\frac{2}{1+3\cos x+4\sin x}$

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    I see, but that isn't the maxima/minima of the actual function itself, or is it? Wouldn't it make more sense to refer to the 'greater' point as the maximum and the 'smaller' as the minimum? – 2017-01-23
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    @Koprulu_Most: There are different types of maxima and minima. The ones in these examples are called **local** (or _relative_, but I don't like this name) maxima and minima. What you're asking about are **absolute** maxima and minima, and both of these examples don't have those, exactly because there are no absolutely highest/lowest points on their graphs. – 2017-01-23
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    zipirovich - how can I tell in general when the local maxima will be below the local minima? I am trying to calculate the maxima/minima by using the harmonic form as in my original post, with the logic that f(X) will be maximum when sin(t+36.9) is -1, and minimum when sin(t+36.9) = 1, but that doesn't still make sense as -0.5 < 0.3333. – 2017-01-25
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    @Koprulu_Most: In general -- we probably can't. We have to examine each given function to see its behavior and its features, such as local or absolute extrema (or anything else we may be interested in). One of the key features of this function is that its graph has vertical asymptotes -- and thus the graph consists of a several disjoint branches. Effectively, they are like completely separate curves, albeit shown on the same picture. So what really doesn't make sense is to compare points on what essentially are separate curves. – 2017-01-25
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HINT: the first derivative is given by $$f'(x)=-2\,{\frac {-3\,\sin \left( \theta \right) +4\,\cos \left( \theta \right) }{ \left( 1+3\,\cos \left( \theta \right) +4\,\sin \left( \theta \right) \right) ^{2}}} $$ solve the equation $$f'(x)=0$$