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Let us suppose I have an unknown function $f(x)$.

I have the information that $f(2)=6$ and $f'(2)=-1$.

What would the actual function be? Are there unlimited possibilities? How do I make an expression that satisfies this information?

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  • $\begingroup$ Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. $\endgroup$ Sep 10, 2021 at 9:44
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    $\begingroup$ You cannot tell what the whole function is by the information you are providing. The simplest way to find one(!) function is to chose $a,b$ properly such that $f(x)=ax+b$ satisfies your conditions. $\endgroup$
    – user592521
    Sep 10, 2021 at 9:44
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    – Community Bot
    Sep 10, 2021 at 9:51

3 Answers 3

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There are multiple fonctions that satisfy this. The "easiest" way is to consider a polynomial function as $$ f\left(x\right)=ax+b $$

You have $$ f\left(2\right)=2a+b=6 $$ and $$ f'\left(2\right)=a=-1 $$ Hence an example is $$ f\left(x\right)=8-x $$

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Yes: there are unlimited options. One of the simplest would be the affine function

$$f(x)=-x+8$$ which goes through the point $(2,6)$ and whose slope is $-1$ (actually everywhere), as requested.

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As it was mentioned in other answers, there are countless possibilities. To obtain a relatively large selection of such functions, you can draw inspiration from Taylor's series, where $$ f(x) = \sum_{n=0}^{+\infty}\frac{f^{(n)}(a)}{n!} (x-a)^n, \quad x \in D. $$

Any function of the form $$ f(x) = 6 - (x-2) + \sum_{n=2}^{+\infty} c_n (x-2)^n $$

is a good fit, as long as the series is convergent. In particular, taking $c_n = 0$, you get $f(x)=6-(x-2)$.

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