Find Absolute Maxima and Minima of f(x) on [-2,2]

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SUMMARY

The discussion focuses on finding the absolute maxima and minima of the function f(x) = (2x)/(x^2 + 1) on the interval [-2, 2]. The derivative of the function is calculated as f'(x) = (2x^2 - 2)/(x^2 + 1)^2. Critical points, where the derivative equals zero, are identified as essential for determining maxima and minima, as they indicate where the derivative changes sign. Resources for further understanding of these concepts are provided, emphasizing the importance of grasping the assignment requirements before attempting the problem.

PREREQUISITES
  • Understanding of calculus concepts, specifically derivatives and critical points
  • Familiarity with the function f(x) = (2x)/(x^2 + 1)
  • Knowledge of how to analyze sign changes in derivatives
  • Basic skills in solving equations for critical points
NEXT STEPS
  • Study the process of finding critical points in calculus
  • Learn about the First Derivative Test for identifying maxima and minima
  • Explore the concept of continuity and differentiability in functions
  • Review applications of derivatives in real-world scenarios
USEFUL FOR

Students studying calculus, educators teaching optimization problems, and anyone interested in understanding the behavior of functions through derivatives.

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1. Find the absolute maxima and the absolute minima of the following function

f(x)=(2x)/(x^2+1) on [-2,2]
 
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i found the derivative = (2x^2-2)/(x^2+1)^2
 
Okay, what will the derivative be at a maximum point?
 
I can't solve it
 
A maximum would occur if the derivative exists and changes sign from positive to negative at a certain point.

A minimum would occur if the derivative exists and changes sign from negative to positive at a certain point.

This means the derivative must pass through zero to change the sign. So, equate your derivative to zero and solve for the x coordinates of these points. In Calculus, these points are called critical points.

Just in case you don't understand, I'd recommend the following resources:

http://ltcconline.net/greenl/courses/115/applications/frsttst.htm
http://www.math.wvu.edu/~hjlai/Teaching/Tip-Pdf/Tip1-21.pdf
http://www.math.ucdavis.edu/~xiaoh/16a/extrema.pdf

It's important to understand what the assignment is on before attempting it.
 
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