Increasing/Decreasing function, no max/min

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Discussion Overview

The discussion revolves around the behavior of a rational function, specifically focusing on its increasing and decreasing nature, and the absence of maximum or minimum values. Participants analyze the critical points and the implications of vertical asymptotes on the function's graph.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant calculates the derivative f ' (x) and identifies critical numbers at 3 and -3, questioning whether these are real roots or complex roots.
  • Another participant clarifies that the expression for -(x^2) - 9 has complex roots, while the expression +(x^2) - 9 has real roots at 3 and -3.
  • A different participant argues that the rational function has vertical asymptotes at x= 3 and x= -3, explaining the behavior of the function in various intervals and asserting that there are no turning points, maximum, or minimum values.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of critical points and the function's behavior, indicating that the discussion remains unresolved regarding the implications of these critical points and the overall characteristics of the function.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the nature of the critical points and the behavior of the function near the vertical asymptotes. The dependence on the definitions of increasing and decreasing functions is also present.

AquaGlass
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thanks!
 
Last edited:
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I obtained f ' (x) = [-(x^2) - 9]/[[(x^2) - 9]^2]
I found the critical numbers of 3 and -3.
Are these real roots, or did you mean 3i, -3i?
 
+(x^2) - 9 has real roots 3 and -3.

-(x^2) - 9 = -((x^2) + 9) has the complex roots I stated.
 
I'm wondering why you find it hard to believe this has no max or min. This is a rational function that has vertical asymptotes at x= 3 and x= -3 (because the denominator factors as (x-3)(x+3)). If x< -3, all of x, x-3, and x-3 are negative so the fraction itself is negative. The graph goes from asymptotic to 0 for large negative x to going to -infinity as it approaches x= -3. If -3< x< 0, the fraction is positive. The graph goes from +infinity close to x= -3 through 0. If 0< x< 3, the fraction is negative and goes from 0 toward - infinity as x approaches 3. Finally, for x> 3 the fraction is negative and runs from near + infinity close to 3 to asymptotic to 0 as x goes to +infinity.
There are no turning points and no max or min.
 

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