What is the largest absolute value attained by the function?

  • MHB
  • Thread starter Ackbach
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    2015
In summary, the largest absolute value attained by a function is the greatest numerical value that the function can reach, regardless of whether it is positive or negative. It can be found by graphing the function or using calculus to find the critical points. Yes, the largest absolute value can be negative and not all functions have a largest absolute value. It is important because it provides information about the function's behavior and range, and can help determine the maximum or minimum value in real-world applications.
  • #1
Ackbach
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MHB
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Here is this week's POTW:

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What is the largest absolute value attained by the function $f(z)=z^{2000}-z^4+1$ as $z$ ranges over the unit circle in the complex plane?

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
Congratulations to Sudharaka for his correct solution to this week's POTW. His solution follows:

By the triangle inequality,

\[|z^{2000}-z^4+1|\leq |z^{2000}|+|z^4|+1\]

Since $z$ is on the unit circle, $|z|=1$. Therefore,

\[|z^{2000}-z^4+1|\leq 3\]

Notice that the maximum value of f(z) is achieved when $z=\sqrt[4]{-1}$.

\[|f(\sqrt[4]{-1})|=|(\sqrt[4]{-1})^{2000}-(\sqrt[4]{-1})^4+1|=|(-1)^{500}-(-1)+1|=3\]
 

FAQ: What is the largest absolute value attained by the function?

What is the largest absolute value attained by the function?

The largest absolute value attained by a function is the greatest numerical value that the function can reach, regardless of whether it is positive or negative.

How do you find the largest absolute value attained by a function?

To find the largest absolute value attained by a function, you can graph the function and visually determine the maximum absolute value. Alternatively, you can use calculus to find the critical points of the function and determine the absolute value at those points.

Can the largest absolute value attained by a function be negative?

Yes, the largest absolute value attained by a function can be negative. The absolute value of a negative number is the positive version of that number. Therefore, the largest absolute value attained by a function can be a negative number that has the greatest magnitude compared to all other values the function can attain.

Does every function have a largest absolute value?

No, not every function has a largest absolute value. Some functions may continue to increase or decrease without ever reaching a maximum absolute value.

Why is the largest absolute value attained by a function important?

The largest absolute value attained by a function is important because it gives us information about the behavior and range of the function. It can also help us determine the maximum or minimum value of a function, which can be useful in various real-world applications.

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