Consider the functional
\(\rm I(y)=\int_a^b F\left(y, y^{\prime}\right) d x \); \(\rm y^{\prime} \equiv \frac{d y}{d x}\)
y(a) = y1, y(b) = y2
where y ∈ C2 [a, b], F has second order continuous partial derivatives with respect to y, y', and y1, y2 are given real numbers. Let y = y ( x ) be an extremizing function for the functional I . Then, along the extremizing curve
1
F remains constant
2
\( \frac{\partial F}{\partial y}=0\)
3
\( F-y \frac{\partial F}{\partial y^{\prime}}=\text { constant }\)
4
\( F-y^{\prime} \frac{\partial F}{\partial y^{\prime}}=\text { constant }\)
5
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