need to find the Largest and Smallest values of a Trig func.

[lalala84]

Hi,

I need to find the largest and smallest values of:
f(x) = 5 - 3cos2x

I know I need to use inequlaties, but I'm stuck as to how to begin the question.

I started it like this: - 3cos2x < -5 .....when I looked in the solution manual it had something totally differerent. I'm really stuck and any help to help me get started in the right direction would be appreciated.

The answers are 2 for the min. value & 8 for the max. value. I have the solution, but I don't understand it.

Thanks for any help in advance.

Kalli

p.s. is this posted in the right section? I'm in a calculus course and this is from the trig section that i'm in.

 

[pka]

\(\displaystyle \L
\begin{array}{l}
- 1 \le \cos (2x) \le 1 \\
- 3 \le - 3\cos (2x) \le 3 \\
2 \le 5 - 3\cos (2x) \le 8 \\
\end{array}\)

 

[lalala84]

\(\displaystyle \L
\begin{array}{l}
- 1 \le \cos (2x) \le 1 \\
- 3 \le - 3\cos (2x) \le 3 \\
2 \le 5 - 3\cos (2x) \le 8 \\
\end{array}\)

That's exactly what was in the solution manual, but I don't understand it. Could you please explain it to me? Like, why do you choose - 1 and 1, and then -3 and 3.
:?: Is there a way to do it without inequalities?

 

[pka]

Is there a way to do it without inequalities?

Absolutely Not! There is no other way!

\(\displaystyle - 1 \le \cos (\theta ) \le 1\): this is a basic property of sine and cosine functions.

Now multiply by –3, \(\displaystyle 3 \ge - 3\cos (2x) \ge - 3.\)

Now add 5 to each member of the inequality.

 

[lalala84]

Is there a way to do it without inequalities?

Absolutely Not! There is no other way!

\(\displaystyle - 1 \le \cos (\theta ) \le 1\): this is a basic property of sine and cosine functions.

Now multiply by –3, \(\displaystyle 3 \ge - 3\cos (2x) \ge - 3.\)

Now add 5 to each member of the inequality.

Thanks so much!

So, from the basic property, I just "manipulated" it to be like the equation given in the question? Sorry, If that's a really dumb question. But thanks so much, again

 

[tkhunny]

It was manipulated AND the manipulator paid attention to the implications of the various manipulations. Just playing with it will not do.

 

[lalala84]

It was manipulated AND the manipulator paid attention to the implications of the various manipulations. Just playing with it will not do.

oh.... :? okay...

so, the max and min values (8 and 2) would occur when x = o and x= pi/2 anywhere along the cosx function? Is that interpreted correctly?