Max Extrema and Point of Inflection

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SUMMARY

The discussion focuses on finding the maximum extrema and point of inflection for the function g(t) = 1 + (2 + t)e^(-t). The first derivative, g'(t) = -e^(-t)(1 + t), identifies the critical point at t = -1, which corresponds to a maximum at the point (-1, 1 + e). The second derivative, g''(t) = (t)e^(-t), reveals the point of inflection at t = 0, leading to the coordinates (0, 3). The key takeaway is that to determine the actual points, one must evaluate g(t) at the identified critical and inflection points.

PREREQUISITES
  • Understanding of calculus concepts such as derivatives and critical points.
  • Familiarity with exponential functions and their properties.
  • Ability to compute and interpret first and second derivatives.
  • Knowledge of how to evaluate functions at specific points.
NEXT STEPS
  • Study the application of the First Derivative Test for identifying local extrema.
  • Learn about the Second Derivative Test for determining concavity and points of inflection.
  • Explore the properties of exponential functions, particularly in calculus contexts.
  • Practice evaluating functions at critical points to find corresponding y-coordinates.
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Students and professionals in mathematics, particularly those studying calculus, as well as educators looking for examples of extrema and inflection points in functions.

kari82
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1. find the extremas and points of inflections



2. g(t)= 1+(2+t)e^(-t)



3. So i know you need to find g(t)' and g(t)''

g(t)'= -e^(-t)(1+t)
g(t)''= (t)e^(-t)

my critical point is t=-1 (max) and my point of inflection is t=o

How do I get to max extrema being (-1, 1+e) and point of inflection (0,3)? Thank you very much!
 
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You've only given the x-coordinates of your critical point and point of inflection. The actual point is (t, g(t)). In other words you need to plug the values for t you have into the original function to get the corresponding y-coordinate.
 
Thank you so much! I knew it wasnt hard... I just couldn't figure it out. Thanks again!
 

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