analytically finding max and min

[renegade05]

Alright, here is the question:
Analytically find the exact maxiumum and minimum value for
\(\displaystyle f(x)=\sqrt{3}x-cos(2x)\) on the interval \(\displaystyle [-\pi,\pi]\)

I found through differentiating and solving for x when \(\displaystyle f'(x)=0\) that:

QUAD IV: \(\displaystyle x=\frac{5\pi}{6} + \pi K\)
QUAD III: \(\displaystyle x=\frac{2\pi}{3} + \pi K\)

Where K is integer.

My question is, how can I find what values to put in for K so that the equations remain on the interval \(\displaystyle [-\pi,\pi]\) ?

I mean, I plugged everything into my calculator and it looks i get the critical numbers when k = -1 and 0. But how would i find these values for K analytically.

I am sure i am forgetting something fundamental or easy. Please help. Thanks.

 

[tkhunny]

You seem to be missing the point of the "K". You "put in" ALL POSSIBLE values of "K" and pick the ones that land in the desired Domain.

Of course, you also need to check the endpoints.

 

[renegade05]

so its just trail and error?

 

[tkhunny]

No, it's an understanding of the function and the notation and the operation.

K = 0: Is 2pi/3 in [-pi,pi]? Yes. Great. Local Min or local max? Global or not?
K = 1: Is 5pi/3 in [-pi,pi]? No. Skip it.
K = 2: Why would we try that?
K = -1: Is -pi/3 in [-pi,pi]? Yes. Great. Local Min or local max? Global or not?
K = -2: Is -4pi/3 in [-pi,pi]? No.

Now, move on to the next solution.

 

[renegade05]

Hey what is the proper notation to say: K is set of all integers ?

\(\displaystyle K \epsilon \mathbb{Z}\) or something?

 

[tkhunny]

Integers. The notation \(\displaystyle k\pi\) is intended to suggest "integer multiples of \(\displaystyle \pi\)."

It should be noted that \(\displaystyle \pi\) is the period of the function. If the period were \(\displaystyle 2\pi\), it would be more appropriate to write \(\displaystyle k(2\pi)\) or \(\displaystyle 2k\pi\) and retain the integer definition of k.