The functional $I[y(x)]=\int_{0}^{2}(xy^{'}+y^{'2})dx,y(0)=1,y(2)=0$ possess
a.strong minima
b.strong maxima
c.strong maxima but not weak minima
d.weak maxima but not strong minima
How do we show if the functiona is strong minima,maxima..how do we prove this? what all we have to check?
It has Strong minima.
Legendre Condition: $F_{y'y'}$ (differentiate $F$ w.r.t $y'$ twice) $>0$ for every $y$ then strong minima and if$ F_{y'y'}<0$ for every $y'$ then Strong maxima.
for $>0$ for some $y'$ close to $p=(dy/dx, y\text{ is extremal})$ weak minima and for $<0$ for some $y'$ close to $p=(dy/dx, y\text{ is extremal})$ weak maxima