Myster Curev: Finding the points on the function

[ilovemynny]

The following five points lie on a function: (1, 20), (2, 4), (5, 3), (6, 2), (10, 1). Find a function that passes through these points and has these features:

1. There are three inflection points
2. There is at least one relative maximum
3. There is at least one relative minimum
4. At least one of your critical numbers does NOT correspond to any of the given points.
5. The curve is continuous and differentiable throughout
6. The function is not a single polynomial, but must be a piecewise-defined function

I know how to find the relative max and min (pretty sure), I just don't know how to make a piece wise function for this. I've tried it a couple of times, and it's not working out. Since it asks that the curve is continuous this would mean that x can't have different values so every graph I make up is very useless.

I hope someone can help me or at least explain to me how i can get this started

 

[HallsofIvy]

First, this has nothing to do with "differential equations" so I am moving it to "Calculus and beyond Homework".

It looks to me like a "piece-wise" function would be simpler than a polynomial.

Just break the problem into parts- say, from x= 1 to 5 for one part, x= 5 to 10 for the second part. Find one polynomial satifying the conditions at x= 1, 2, and 5, then another satisfying the conditions at x= 5, 6, and 10.