Finding extrema on a graph of trigonometric functions?

Question

There are two difficulties I'm having. First, the problem itself:

For the following problem, (a) Find the intervals on which f is increasing or decreasing, and (b) Find the local maximum and minimum values of f.

14. f(x) = cos^2(x) - 2*sin(x), 0 =< x =< 2*pi

I've come up with the following critical points: 0, pi/2, 4 - 5(??), and 2*pi. First of all, I don't know what's up with the 4 - 5 interval, I mostly discovered it by graphing, though I think something led me to examine it initially. It seems to hold the absolute maximum. How would I actually analyze and find information about that interval? Secondly, how does one fully analyze a trigonometric function? I see that they put it on an interval, so I assume that one should just work through cases? Obviously at this point I can't just algebraically solve the equation for zero.

Any help is appreciated! Thanks!

Thank you very much -- a little bit after I posted this, I realized that I was looking at 3*pi/2. I just wasn't thinking "periodically" right off the bat. Anyhow, I've solved the problem now, so thank you both!

Answer 1

I tried to graph this, and it would be helpful to graph both parts of f(x) separately. First, 0 and 2pi should yeild the same result for the function.
The cos^2(x) is a periodic function, but it varies from 0 to 1 in one period. The 2sin(x) is a periodic function, that varies from -2 to +2 in one period. Both have one period in the 0 to 2pi interval of x.

Then you can note that f(x) will decrease from +1 at x=0 to -1 at pi/2. Then it will increase to +1 at pi, and further to +2 at 3pi/2. Then it will decrease to +1 at 2 pi. If you are taking calculus, you can solve for minimum and maximum (points where the function is "flat"). If not, working through cases is as good an exercise as any, but you should have knowledge that can help you take shortcuts.

Answer 2

a) Decreasing from 0 to Pi/2, increasing to 3*Pi/2, decreasing to 2*Pi.
b) Local min at Pi/2 of -2, max at 3*Pi/2 at 2.

The 4-5 interval you speak of is 3*Pi/2 (or 270 degrees). I graphed it using Excel, and it does seem to hold around that maximum on each side. However, expanding the result (more digits shown to the right of the decimal), you can see the result is very close to the maximum. Keep in mind the cosine is close to zero, and you're squaring it (so even closer), and the sin is close to 1.