Making Piecewise Function Continuous w/ First & Second Derivatives

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SUMMARY

The discussion centers on ensuring a piecewise function is continuous at x=0, specifically addressing the continuity of first and second derivatives. The user successfully equated the right-hand side (RHS) and left-hand side (LHS) limits to find the value of variable 'a', achieving continuity for the first derivative, which is zero for both sides. However, the second derivatives exhibit a discontinuity, with one approaching infinity and the other zero, raising questions about the implications of this discontinuity on the overall function's continuity. The user seeks clarification on the relationship between the variables 'a' and 'b' and the function's formula.

PREREQUISITES
  • Understanding of piecewise functions
  • Knowledge of limits and continuity in calculus
  • Familiarity with first and second derivatives
  • Basic algebra for solving equations
NEXT STEPS
  • Research the implications of discontinuous second derivatives in piecewise functions
  • Study methods for ensuring continuity in piecewise-defined functions
  • Learn about the relationship between parameters in piecewise functions
  • Explore examples of piecewise functions with continuous first derivatives but discontinuous second derivatives
USEFUL FOR

Students and educators in calculus, mathematicians working with piecewise functions, and anyone interested in the continuity of derivatives in mathematical analysis.

Dollydaggerxo
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I have to make a piecewise function continuous with first and second derivatives at x=0 by finding a value for a and b.

I have made the function continuous by equating the RHS and LHS limits and then solving for the variable a.

The limit of the first derivatives of the LHS and RHS are the same, both 0, so that makes the first derivative continuous...

However, the limits of the second derivates for the RHS and LHS are different. One is infinty when the other is 0.

My question is what does this mean exactly? Is the question not possible if my second derivatives are not continuous?
 
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First, please tell us what the problem really is! You say "by finding a value of a and b" but tell us nothing about how "a" and "b" are related to the function. Are you given a formula for the function involving a and b? If so, that formula is important information.
 

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