Can roots be at the same points as POI's?

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Discussion Overview

The discussion revolves around the relationship between roots of a function and points of inflection (POIs). Participants explore whether it is possible for these two features to coincide in a function's graph, with a focus on specific examples and theoretical implications.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions if roots can be at the same points as POIs, noting that both conditions yield the same solution for their equations.
  • Another participant affirms that it is indeed possible, providing the example of the function f(x)=x^3, which has a root and a POI at x=0.
  • A further contribution highlights that plotting Y=X^3 visually demonstrates the change of slope at the point of inflection, reinforcing the previous point.
  • Another participant introduces the idea that vertical shifts of a function can allow for a POI to coincide with a root, suggesting a method to construct such functions.

Areas of Agreement / Disagreement

Participants generally agree that roots can coincide with POIs, with examples provided to support this view. However, the discussion remains open to further exploration of the implications of this relationship.

Contextual Notes

Participants do not delve into the mathematical conditions required for roots and POIs to coincide, nor do they clarify the implications of vertical shifts on the characteristics of the function.

frenkie
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can roots be at the same points as POI's? because when I set my original equation to zero and I set my second derivative equation to zero i get the same answer? is it possible that roots can be same as the POI's.
 
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Yes. Just look at [itex]f(x)=x^3[/itex] in any interval containing [itex]x=0[/itex].
 
Of course, and if you plot Y=X^3, you will see the change of slope at X=0.
 
Something else to ponder:

In your precalculus course you learned that you can shift a function vertically by adding a real number to it. That means that no matter what the y-coordinate of the POI of a function, you can always construct another function whose POI is a root, by doing an appropriate vertical shift.
 

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