Okay, so this is a pretty big problem and I KNOW this is asking for TONS. But we've been working on this for so long and we just can't get it. And it's due tomorrow, and my entire group is freaking out.

five points lie on a function (1,20) (2,4) (5,3) (6,2) (10,1)
there are only 3 inflection points
there is at least one local maximum
there is at least one local minimum
at least one critical point is not at a given point
the curve is continuous and differentiable throughout
the equation is not a single polynomial but must be a piecewise defined function
prove that your answer function meets these criteria

So far we've been using the equation:
From negative infinity to positive two.

But we get stuck after this. We get what we're supposed to do, but it's just so ridiculously hard to get two equations to meet at certain point and be differentiable there.

Any piece of the puzzle would be REALLY appreciated. You don't have to give us it all. Or any simpler way of doing this besides just guessing and checking over and over. We've been working on this problem for hours and we've gotten nowhere.

Oh, and you don't have to be exact with your numbers, as long as the numbers round up / down from the hundreths place. So if you're trying to get a derivative to equal -16 at a point, -15.96 is acceptable.