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

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SUMMARY

Roots can indeed coincide with points of inflection (POIs) in certain functions, as demonstrated by the function f(x) = x^3. When analyzing this function, both the original equation and its second derivative yield the same critical point at x = 0, indicating that a root and a POI can overlap. Additionally, vertical shifts of functions allow for the construction of new functions where the POI becomes a root, further illustrating this relationship.

PREREQUISITES
  • Understanding of calculus concepts, specifically roots and points of inflection.
  • Familiarity with polynomial functions, particularly cubic functions.
  • Knowledge of vertical shifts in function transformations.
  • Ability to analyze derivatives, including first and second derivatives.
NEXT STEPS
  • Study the properties of cubic functions, focusing on their roots and inflection points.
  • Learn about vertical shifts in functions and their effects on roots and POIs.
  • Explore the relationship between first and second derivatives in determining critical points.
  • Investigate other polynomial functions to see if roots and POIs coincide in different cases.
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Students studying calculus, mathematics educators, and anyone interested in the behavior of polynomial functions and their derivatives.

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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 f(x)=x^3 in any interval containing x=0.
 
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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