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Math Help - Max and Min confusion

  1. #1
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    Max and Min confusion

    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?
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  2. #2
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    Quote Originally Posted by Alexrey View Post
    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
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    Would |e^i theta| = 1 so the above modulus would be 1+R?
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    Quote Originally Posted by Alexrey View Post
    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
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  5. #5
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    Sorry, I have no idea what to do next :/
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    Quote Originally Posted by Alexrey View Post
    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
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  7. #7
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    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?
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Alexrey View Post
    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
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  9. #9
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    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?
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  10. #10
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    Quote Originally Posted by Alexrey View Post
    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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