What Function Minimizes Arc Length While Meeting Specific Conditions?

[Mandanesss]

Homework Statement

Find a formula for one function that satisfies these given conditions:

1.) f is continuous on [0,1].
2.) f(0) = f(1) = 0 (exactly).
3.) f(x) is greater than or equal to 0 for [0,1]
4.) The area under the graph of f from 0 to 1 is equal to 1 (exactly).

Calculate the arc length.
Try to find the smallest possible arc length.

Homework Equations

Arc Length formula

The Attempt at a Solution

I don't know! I'm completely lost. So far I found the function f(x) = -6x^2+6x gives me an arc length of ~3.249 and satisfies all four conditions. But I'm pretty sure there must be a smaller arc length. Please help me find the smallest possible arc length. Thank you!

 

[HallsofIvy]

All I can do is point out that f(x)= 6x²+ 6x certainly does not satisfy all the conditions. The "area under the graph of f from 0 to 1" is 5, not 1!

 

[Dick]

All I can do is point out that f(x)= 6x²+ 6x certainly does not satisfy all the conditions. The "area under the graph of f from 0 to 1" is 5, not 1!

The coefficient of the x^2 term is -6. Not 6. So his curve does work. As to the larger problem, it sounds pretty hard. Where did you find it? Are you doing calculus of variations, or might there be some simple trick?

 

[Dick]

You might try fitting a circular arc to your conditions. Does it give smaller arc length?

 

[Dick]

Of course, if you do that, it won't be a function. But still. This is like a minimal surface problem (soap films). I used to study that! Long ago. Must. Remember.

 

[Dick]

You also might want to try fitting a cosh(x) type curve. It's a catenary. Minimizes some other things. It does have the virtue that it MAY BE easy to compute the arc length exactly.

 

[gammamcc]

I don't know at what level you are expected to prove this, but you can pick very simple functions since you are only expected to use continuous functions. Try piecewise-linear functions.

(Euler - Lagrange equations show that dy/dx = const minimizes arc length, but you have an additional constraint that makes this result unsatisfactory.)

 

[Mandanesss]

This question is for Calculus II, we haven't done anything with Euler - Lagrange equations...yet. This question was given to us as class contest (with no prize!).

 

[Dick]

If you haven't done Euler-Lagrange yet, then you probably don't want to plunge into the formulism. But your geometric intuition along might tell you what kind of curve to try.