Use the given data to find an approximation for $f(x)$.

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I have the following question in the book:

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But I do not know how to answer it, the only piece of information that my professor said is that a function differentiable means that it can be approximated by linear function, could anyone explain for me how to answer this question please?

EDIT:

The question contains a mistake as the vector x is 3 dimensional not 2.