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Real Analysis : Derivatives

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If I say that the function f:R->R has two derivatives, with f(0) = f'(0) = 0 and the absolute value of f"(x) is less than or equal to one, if the absolute value of x is less than or equal to 1. How can I prove that:

f(x) <= 1/2 if x <= 1

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Solution Summary

Integration is used to prove a function is true. The absolute value of x being less or equal to one is computed.

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because,
|f''(x)| <= 1 for |x| <= 1
=> -1 <= f''(x) <= 1 for -1<=x<=1
integrate from 0 to x:
=> -x <= f'(x) <= x for -1<=x<=1 (because ...

Solution provided by:
Amrit Lal Ahuja, PhD (IP)
Education
  • BEng, Allahabad University, India
  • MSc , Pune University, India
  • PhD (IP), Pune University, India
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