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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?

Kenta S
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amit
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1 Answers1

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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

Kenta S
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user3057692
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    @user30567692 I tried to edit your answer but the I could not understand the second half please try to use LATEX – happymath Jan 23 '15 at 09:52
  • In your example, Fy'y' = 2>0 for every y'. Therefore it is strong minima. Let us assume thet for some problem your Fy'y' = y'+x, Now its sign is not clear and it depend on sign of p=dy/dx (where y is extremal) and range of x, So it is definitaley week ( maxima or minima will depend on sign) – user3057692 Mar 14 '15 at 16:10