Finding Local Max/Min with f(x)= x + 9/x: Explained

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SUMMARY

The discussion focuses on finding local maxima and minima for the function f(x) = x + 9/x. The user identifies that the function has a local minimum at the point (-3, -6). The intervals where the function is increasing and decreasing are also explored, with the user suggesting that f(x) is decreasing on the interval [-3, 0) and increasing on (0, 3]. The presence of a vertical asymptote at x = 0 is acknowledged, affecting the behavior of the function around this point.

PREREQUISITES
  • Understanding of calculus concepts such as local maxima and minima
  • Familiarity with vertical asymptotes and their implications on function behavior
  • Knowledge of interval notation for expressing increasing and decreasing functions
  • Basic algebra skills for evaluating functions at specific points
NEXT STEPS
  • Study the concept of derivatives to determine increasing and decreasing intervals
  • Learn about vertical asymptotes and their effects on function graphs
  • Explore the use of the first derivative test for local extrema
  • Investigate the behavior of rational functions and their critical points
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Students and educators in calculus, mathematicians analyzing function behavior, and anyone interested in understanding local extrema in rational functions.

wr1015
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ok so in the problem f(x)= x + 9/x, I know how to write where f is increasing, but how do you write how it's decreasing? Is it written as [-3,0)U(0,3]? since x=0 is a vertical asymptote wouldn't all the x-values between -3 and 3 be as close to 0 as possible?
 
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wouldnt f(-3) = -3 + 9 / (-3) = -3 -3 = -6
 
mathmike said:
wouldnt f(-3) = -3 + 9 / (-3) = -3 -3 = -6

yeah that's my local min at (-3,-6).
 

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