Min. problem(2)

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圖片參考：http://imgcld.yimg.com/8/n/HA05107138/o/20131006104222.jpg


2013-10-07 20:21:54 補充：
1. Thank you for advising the Euler equation method. But apparently this does not work. Should the question specifies the boundary values, i.e. f(0) and f(1) instead of the initial values f(0) and f'(0) in order to use this method?
2. I have not confused (r,θ) with (x,y).

2013-10-07 20:27:00 補充：
By using Euler equation and the initial values, I can get f(x) = 1/(cos x + sin x) which corresponds to a straight line in my polar coordinates equivalence r(θ) = 1/ (cos θ + sin θ). The integral yields 2 sin (1/2) = 0.9589..
But consider f(x) = 1 - x +x^2/2 the integral = 0.8495 smaller than above

2013-10-07 20:28:59 補充：
My piecewise smooth curve gives even a smaller value of sin(1)=0.8415

2013-10-08 20:38:52 補充：
I think Euler equation requires the 2 boundary conditions f(0) and f(1) to get the correct result. Let's say if we specify f(0)=1 and f(1) = 1, we will get f(x) = 1/(cos x + (1 - cos 1)/sin 1 * sin x) and the integral becomes 2 sin (1/2) = 0.9589 (which I mistakenly copied as the result for

2013-10-08 20:41:40 補充：
f(x) = 1/(cos x + sin x)
This 2 sin (1/2) agrees with the length of straight line from (r,θ) = (1,0) t0 (1,1).
I have tried different values of f(1) and the result is always correct.
Likewise for the integral sort(1+f'(x)^2) we get the same conclusion.

Take it easy, maybe you are confusing (r,θ) with (x,y). Or you can solve by the Euler equation.

2013-10-06 13:04:22 補充：
To: 001
Diff. equation is an equation that contains unknown function(s) and their derivative function(s) in the equation.

2013-10-08 02:01:50 補充：
1. By using Euler eq. we can obtain f(x) = 1/ (cos x + sin x), then
P(f)= sqrt(2) sin1/(sin1+cos1)~0.8612264
2. If g(x)=1-x+x^2/2, P(g)~0.8495
3. If h(x)=1-x-x^2/2, P(h)~1.61146
P(h) > P(f) > P(g) , the Euler eq. gets a wrong sol. why?

2013-10-13 23:53:40 補充：
I took a trip to see bamboo (bamboo sea), now I am back!
I think the solution of Euler method is just a local extreme, not necessarily the global extreme.

diff. equation 是甚麼意思？