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Differentiability

If \(f(x)\) is a polynomial satisfying relation \(f(x)=\dfrac{k}{k+1}\) for \({k=1,2,3,\ldots,100}\) except \(k=a\) where \(a\) is a natural numbers and belongs to \((0,100)\) and \(f(101)=1\).Find \(\dfrac{1-f(a)}{f'(a)}\) .

Note by Akshay Sharma
8 months, 3 weeks ago

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Try reading the first section of polynomial interpolation - remainder factor theorem.

Addendum: Oh wait... this problem is not as easy as it looks. Give me one moment to think about this... Pi Han Goh · 8 months, 3 weeks ago

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Several issues with your problem

  1. There are multiple polynomials that satisfy the conditions since we can add \( A (x-1)(x-2)\ldots (x-101) \). This suggests that you want the degree to be 100.
  2. The condition should be \( f(k) = \frac{k}{k+1} \) instead?
Calvin Lin Staff · 8 months, 3 weeks ago

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