Equation with 3 Inflection Points & Local Extrema | Help Needed!

[HGM915]

Homework Statement

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

a. There are three inflection points
b. There is at least one local maximum
c. there is at least one local minimum
d. at least one critical point is not at a given point
e. THE CURVE IS CONTINUOUS AND DIFFERENTIABLE THROUGHOUT
f. the equation is not a single polynomial, but must be a piecewise defined function

The easiest thing we've tried is to put the cubic parts of the function outside of points from 1 to 10 but we can't seem to make it differentiable. We've tried everything that we could think of... PLEASE help, as this is due tomorrow and we've exhausted all of our options!

Homework Equations

The Attempt at a Solution

I know that the slope at the connecting points of each part of the piecewise function must be equal, but I can't figure out how to make that happen/work. I've tried a variety of linear equations, but those are not differentiable. I'm really stuck, and have been working on this problem for, literally, DAYS. Please help in any way you can!- with either the equation, or tips/advice/help for how to make the derivatives equal without messing around with the whole function. THANKS!

 

[NoMoreExams]

Can't you fit a 4th degree polynomial through those points and then "attach" other functions to the ends?

 

[HGM915]

How would I do that? And how would I be able to make the function continuous and differentiable?

 

[NoMoreExams]

You can fit it using splines or solving simult. equations or using Excel...

 

[Vanadium 50]

How would I do that? And how would I be able to make the function continuous and differentiable?

To make it continuous, you need to match the value of the function at the ends.

To make it differentiable, think about the derivatives at the ends.

 

[HGM915]

I've figured out how to make it continuous and fit the other criteria, but the differentiability is killing me. I just spent about an hour thinking of all the possible equations with the derivatives to no avail. How do I even come up with the equations so they're differentiable? And does anyone at least have an example of an equation that could work so I can try to see how to make it? I have a test tomorrow and haven't begun to understand this...

P.S Thank you all for the help you've already given... I really appreciate it! (I just wish I was a bit brighter so I could comprehend how to do this stuff!)

 

[NoMoreExams]

To be differentiable you just want to make sure that the derivative is continuous so make sure there's no "sharp points". Think about why |x| isn't differentiable.

 

[NoMoreExams]

So how about this figure out what the derivative would be of your fitted polynomial and come up with another one that "joins" it and has the same derivative? (at the endpoints that is)

 

[HGM915]

How would I go about that?

 

[NoMoreExams]

After you fit the curve, figure out what the slope is at the endpoints?

 

[HGM915]

But how would I be able to make a new equation fitting the other endpoint where the derivative is equal to the previous piece?

 

[NoMoreExams]

Can you draw a line that has the same slope? You will have a point that you know will lie on that line (the endpoint) and the slope.