Simpson's Rule : Parabolas passing through three points on a curve While trying to understand the Simpson's rule, I read that a typical parabola passes through three points on a curve. I'm not sure if that's always true. I can imagine a parabola going through two points on a curve. But going through three points is somewhat hard to imagine. 
Let us take the function $f(x)=\frac{e^x}{x}$. Is there a way to know that $ax^{2}+bx+c$ "surely" intersects $f$ for some $(x_1, x_2, x_3)$? 
I could take any set of $x$ values to start checking. But without the knowledge of $a, b, c$, I'm going nowhere, I think. 

 A: Setting Simpson's Rule aside for the moment:

...I read that a typical parabola passes through three points on a curve. I'm not sure if that's always true. I can imagine a parabola going through two points on a curve. But going through three points is somewhat hard to imagine.

Give me three points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ whose $x$-coordinates are distinct, and I can solve for $a,b,c$ such that $y=ax^2 + bx + c$ passes through all three points exactly.  Now this might not be a parabola (since $a=0$ is possible), and the most general parabola might have a rotated axis (not parallel to the $y$-axis as with the curve envisioned here).
It is not hard to see that $a,b,c$ are the solution to a linear system of three equations in those unknowns, based on plugging in each points coordinates $x_i,y_i$ for $i=1,2,3$.  The distinctness of the $x_i$'s guarantees the full rank of this system's coefficients and thus the existence and uniqueness of the solution.
A: For that part of the proof, it's irrelevant that the three points are on a curve.  Given any three points (not in a straight line) you can find a parabola $y=ax^2+bx+c$ that passes through them.  Then, if it happens that the three points are on a particular curve, this parabola approximates the curve.  (Just like the line through two points would approximate the curve.)
