Question

Increment. Help.

Answer 1

\[\int\limits_{0}^{1}x'(t)^{3}dt\]
\[h \in C ^{1}[0,1], h(0)=h(1)=0\]
\[||h||_{1}=\max {|h(t)|} +\max{|h'(t)|}, t \in [0,1]\]
What should \[||h||_{1}\]
be to satisfy
\[J(x(t)+h(t))-J(x(t))\ge 0 ?\]

Answer 2

I've got that\[J(x(t)+h(t))-J(x(t)) = 3\int\limits_{0}^{1}h'^{2}(1+\frac{ 1 }{ 3 }h')dt\]
and then I have solution that \[J(x(t)+h(t))-J(x(t))\ge 0\] if \[if ||h||_1\le 3\], but I don't know why 3.