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MATH 402 - Assignment #2Due on Monday February 7, 2011Name —————————————–Student number —————————
1Problem 1:We say that the functionf(x1, . . . , xn) is positive-homogenous of degreekinx1, . . . , xniff(λx1, . . . , λxn) =λkf(x1, . . . , xn)for everyλ >0. Prove that iff(x1, . . . , xn) is continuously differen-tiable and positive-homogenous of degreek, thennXi=1∂f∂xixi=kf.
2Problem 2:Derive the differential equation which must be satisfied by the functionwhich extremizes the integralI=Zbaf(x, y, y0, y00)dxwith respect to twice-differentiable functionsy=y(x) for whichJ=Zbag(x, y, y0, y00)dxpossesses a given prescribed value, and withyandy0both prescribedat the end pointsaandb.(a) Show that leavingyunspecified at either end point leads to the
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Math, Calculus, Continuous function, Calculus of variations, end point