The problem asks us to prove the length of a curve decreases as one of its parameters, t, increases. Here is the full statement
[tex]L_t=\int_0^1 ||\partial_s x(t,s)||ds[/tex] is the length of the curve as a function of t, for all t greater than 0.
[tex]\partial_t L_t=\int_0^1 \partial_t ||\partial_s x(t,s)||ds[/tex] is the derivative of the length of the curve with respect to t.
The Attempt at a Solution
To show that the length L(t) decreases as t increases, I think we have to show that the derivative of the length L(t) with respect to t is negative, that is, [tex]\partial_t L_t=\int_0^1 \partial_t ||\partial_s x(t,s)||ds < 0[/tex] is negative.
The only strategy I have for the moment is the mean-value theorem which I've seen used to prove inequalities before (e.g. show that sin(x)<x for all x) but so far, I'm still a bit unclear on how to apply it to this situation. The other attempt I've tried was to take the hint, do some substitutions and hope something will pop out but for a 3 mark question, there's no way this method alone will work. I'm also not yet able to see how the information at the start is relevant to doing this problem.
Any helpful pointers will be greatly appreciated, thanks!
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