Max and Min confusion

**Alexrey** · May 14th 2011, 12:59 PM

Hey guys, I have a problem. I'm a little confused on how I go about doing something like this:

Which I was told was the same as:

How do I actually find a max or min like those? Do I just test values until I think I've found the lowest one, or is there a more structured way of doing this?

**topsquark** · May 14th 2011, 01:14 PM

Originally Posted by Alexrey

Hey guys, I have a problem. I'm a little confused on how I go about doing something like this:

Which I was told was the same as:

How do I actually find a max or min like those? Do I just test values until I think I've found the lowest one, or is there a more structured way of doing this?

Since R is constant you can just take the derivative and set it to 0. However it might be a bit easier if you find the modulus first. Note that $e^{i \theta} = cos(\theta) + i~sin(\theta)$ . So what is the modulus of $1 + Re^{i \theta}$ ?

-Dan

**Alexrey** · May 14th 2011, 01:27 PM

Would |e^i theta| = 1 so the above modulus would be 1+R?

**topsquark** · May 14th 2011, 01:32 PM

Originally Posted by Alexrey

Would |e^i theta| = 1 so the above modulus would be 1+R?

|a + b| is not equal to |a| + |b|.
$|1 + Re^{i \theta}| = |1 + R(cos(\theta) + i~sin(\theta))| = |(1 + R~cos(\theta)) + i~R~sin(\theta)|$

Can you go from there?

-Dan

**Alexrey** · May 14th 2011, 01:37 PM

Sorry, I have no idea what to do next :/

**topsquark** · May 14th 2011, 04:31 PM

Originally Posted by Alexrey

Sorry, I have no idea what to do next :/

By definition:
$|a + i~b| = \sqrt{(a - i~b) (a + i~b)} = \sqrt{a^2 + b^2}$

-Dan

**Alexrey** · May 14th 2011, 10:42 PM

I can find the max and min of cos(x) and sin(x) on their own but together poses a challenge for me. Apologies again for my stupidity, but how is the above definition going to help me to find a max or min?

**topsquark** · May 15th 2011, 12:25 AM

Originally Posted by Alexrey

I can find the max and min of cos(x) and sin(x) on their own but together poses a challenge for me. Apologies again for my stupidity, but how is the above definition going to help me to find a max or min?

C'mon. If you are at this level you are betting than this. Apply yourself! To find a max or min you take the derivative and set it equal to 0. As for the function:
$|1 + Re^{i \theta}| = |(1 + R~cos(\theta)) + i~R~sin(\theta)| = \sqrt{(1 + R~cos(\theta))^2 + (R~sin(\theta))^2}$

$= \sqrt{1 + 2R~cos(\theta) + R^2~cos^2(\theta) + R^2~sin^2(\theta)} = \sqrt{1 + 2R~cos(\theta) + R^2}$

So, for example, you are trying to find the minimum of $|1 + Re^{i \theta}|$ :
$\frac{d}{d \theta}|1 + Re^{i \theta}| = - \frac{R~sin(\theta)}{\sqrt{1 + 2R~cos(\theta) + R^2}}$

Setting this equal to 0 gives $\theta = n \pi$ or R = 0. (If all you care about is the variation of (theta) then ignore the R = 0 possibility, though it really can potentially affect the final answer.)

Use the second derivative test to decide if n is even or odd. The proof using the maximum is not much more complicated.

-Dan

**Alexrey** · May 18th 2011, 08:19 AM

I asked my lecturer and he said that there was is no need to find max/min using derivatives, he said that it is all to do with simple estimation. Is there a way to find the max/mins of the above by estimation?

**HallsofIvy** · May 18th 2011, 08:44 AM

Originally Posted by Alexrey

Hey guys, I have a problem. I'm a little confused on how I go about doing something like this:

Which I was told was the same as:

How do I actually find a max or min like those? Do I just test values until I think I've found the lowest one, or is there a more structured way of doing this?

$1+ Re^{i\theta}$ , as $\theta$ goes from 0 to $2\pi$ , is a circle with center at 1= (1, 0) in the complex plane and radius R. The max and min will be at the points closest to and farthest from 0= (0, 0). And those will lie on the straight line through (0, 0) and (1, 0).

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