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I need to prove that a quadratic spline $S$ and a quadratic function $f$ are equal.

We are given points $([X_1,f(X_1)],.....,[X_n,f(X_n)])$ and the boundary condition that the first derivatives of $S(x_1)$ and $f(x_1)$ are equal.

Obviously that's true but how should one prove it? Thanks in advance!

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    1. Do you mean "A quadratic spline is a _piecewise_ quadratic function", or are there only three points? 2. Do you know the respective definitions? Where are you stuck? <> On a larger-scale note, this site is intended to be an archive of questions and answers, not (for example) a site where students come to get others to do their homework. It's appreciated if you give context to your questions (what do you know, what have you tried), and if you upvote and select the answer(s) that helped you most. The [site tour](http://math.stackexchange.com/tour) is worth a visit. [...] – 2017-01-11
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    If your questions _are_ homework, that's fine, but please say so, and cite where the question comes from. To do otherwise is dishonest, both to the person who created the question and to potential respondents. – 2017-01-11
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    Yes, this is a question from a homework assignment. I have struggled with that question, that is the reason why i am asking for help. I have been stuck on this problem for some time. Answering your question: There are a random number of points (Lets say n, where n>= 2) in the form of [Xn,f(Xn)]. – 2017-01-11
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    And is by chance the function $f$ itself the in the title mentioned quadratic function? I.e., it is to show that the piecewise quadratic approximation of a quadratic function is exact if the given initial condition is satisfied? – 2017-01-11
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    Yes, actually your assumption is correct. – 2017-01-11

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