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 $\dfrac{1-f(a)}{f'(a)}$ .

#Calculus

Easy Math Editor

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Comments

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 - 5 years, 3 months ago

Several issues with your problem

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.
The condition should be $f(k) = \frac{k}{k+1}$ instead?

Calvin Lin Staff - 5 years, 3 months ago

Several issues with your problem

There are multiple polynomials that satisfy the conditions since we can add . This suggests that you want the degree to be 100. The condition should be instead?

Nahom Assefa - 2 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$