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Let $$f(z)=(z -1)(z -4)^{2}$$ Find the lines (through $z=2$) on which $|f(z)|$ has a relative maximum, and the ones on which $|f(z)|$ has a relative minimum.

MY ATTEMPT:

"The line z= 2" is the line where z= 2+ ix for any real number x. Then $f(z)=(z−1)(z−4)^{2} =(2−ix−1)(2−ix−4)^{2}=(1−ix)(−2−ix)^{2}$ . Write out $|f(z)|$ as a function of x, find the derivative, set it equal to 0.

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What have you tried so far? –  Brett Frankel May 1 '12 at 20:17
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I'm confused by the usage of the phrase "through $z=2$" here. What does that mean? –  Thomas Andrews May 1 '12 at 20:26
    
"The line z= 2" is the line where z= 2+ ix for any real number x. Then $f(z)=(z−1)(z−4)^{2} =(2−ix−1)(2−ix−4)^{2}=(1−ix)(−2−ix)^{2}$ . Write out $|f(z)|$ as a function of x, find the derivative, set it equal to 0. –  Breton May 1 '12 at 22:08
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1 Answer

up vote 3 down vote accepted

Here is a hint to get you started. Please ask if you require clarification. If you edit your question to include what you have tried so far I would be more willing to give extra help.

You can parameteize any line going through $z=2$ by

$$ z(t) = 2 + te^{i\theta}, $$

where $\theta$ is the (fixed) angle the line makes with the positive real axis and $t$ is allowed to vary over $\mathbb{R}$. Substitute this into $|f(z)|$ and differentiate with respect to $t$.

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"The line z= 2" is the line where z= 2+ ix for any real number x. Then $f(z)=(z−1)(z−4)^{2} =(2−ix−1)(2−ix−4)^{2}=(1−ix)(−2−ix)^{2}$ . Write out $|f(z)|$ as a function of x, find the derivative, set it equal to 0. –  Breton May 1 '12 at 22:05
    
What then I have to do? –  Breton May 1 '12 at 22:09
    
@Breton, that doesn't make sense. Instead of "The line $z=2$" perhaps you mean $x=2$? And, the question asks about all lines passing through $z=2$, not just the one for $\theta = \pi/2$. Does my answer help? What is unclear? –  Antonio Vargas May 1 '12 at 22:15
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