Discontinuous function at second derivative

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SUMMARY

The discussion centers on the discontinuity of the second derivative of a function at t=1, despite the first derivative being continuous. The user analyzed the function graphically, noting that while both the function and its first derivative are continuous at t=1, the first derivative exhibits a cusp at this point. This cusp leads to a discontinuity in the second derivative, which is confirmed by the differing values of the second derivative from the left (2) and right (0) at t=1.

PREREQUISITES
  • Understanding of calculus concepts, specifically derivatives
  • Familiarity with the graphical representation of functions and their derivatives
  • Knowledge of continuity and discontinuity in mathematical functions
  • Ability to identify cusps and their implications on derivatives
NEXT STEPS
  • Study the concept of cusps in calculus and their effects on derivatives
  • Learn about the criteria for continuity and differentiability of functions
  • Explore graphical methods for analyzing functions and their derivatives
  • Investigate the implications of discontinuities in higher-order derivatives
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in understanding the behavior of functions and their derivatives, particularly in relation to continuity and differentiability.

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Homework Statement



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Homework Equations





The Attempt at a Solution



I was wondering why the second derivative at t=1 does not exist but exists at the first derivative. What I did was draw the graph of the function, then the first derivative and lastly the second derivative. I found that at t=1, the function and its first derivative agreed on both the left hand and right hand sides. However, at t=1 for the second derivative, the top part of the function was 2 and the bottom was 0. Is this why it's not continuous for t=1 at the second derivative?
 
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The function and its first derivative are both continuous at t = 1, but the first derivative has a cusp at t = 1, so the second derivative does not exist at t = 1 .
 

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