# 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
2 years, 4 months 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...

- 2 years, 4 months ago

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?

Staff - 2 years, 4 months ago