Derivatives Affecting Shape of the Graph

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SUMMARY

The discussion centers on analyzing the function h(x) = (x^2 - 1)^3 to determine its intervals of increase and decrease, local extrema, concavity, and inflection points. Participants emphasize the importance of finding the first derivative, h'(x), to identify where the function is increasing or decreasing. The conversation also highlights the necessity of understanding the implications of a function's increasing behavior and how to ascertain these intervals through critical points derived from the derivative.

PREREQUISITES
  • Understanding of calculus concepts, specifically derivatives
  • Familiarity with the first derivative test for increasing and decreasing functions
  • Knowledge of concavity and inflection points in graph analysis
  • Ability to sketch graphs based on derivative information
NEXT STEPS
  • Learn how to compute the first derivative of polynomial functions
  • Study the first derivative test for identifying local maxima and minima
  • Explore the second derivative test for determining concavity and inflection points
  • Practice sketching graphs based on derivative analysis
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in understanding graph behavior through derivatives.

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I have the following question:

h(x) = (x^2-1)^3

a) Find the intervals of increase or decrease
b) Find the local maximum and minimum values
c) Find the intervals of concavity and the inflection points
d) Use the informatin from parts (a)-(c) to sketch the graph

Now I figure a good way of starting this question is just to graph it to begin with anyways. However, the first step really is to find the derivative (f') of this function if I am not mistaken. However, can anyone explain to me how you would go about finding the intervals of increase or decrease for this function. From there, I can figure out the rest. That's all, thanks guys.
 
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Well, what does it mean that a function is increasing?
And how may you ascertain where a differentiable function is increasing?
 

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