Considere a função f(x, y)=3 x^2-5 y^2+7 x y+2 x-4 y-3 :
(a) Calcule partial fpartial x(x, y) e partial fpartial y(x, y)
(b) Calcule partial fpartial x(1,1) e partial fpartial y(-1,0)
(c) Calcule partial^2 fpartial x^2(x, y), partial^2 fpartial y^2(x, y), partial^2 fpartial x partial y(x, y) e partial^2 fpartial y partial x(x, y)
(d) Calcule partial^2 fpartial x^2(0,0), partial^2 fpartial y^2(0,-1), partial^2 fpartial x partial y(1,1) e partial^2 fpartial y partial x(-1,1)

Passo 1

O enunciado apresenta a função
f(x,y) = 3x^2 - 5y^2 + 7xy + 2x - 4y - 3
e pede derivadas parciais de primeira e segunda ordem. Vamos usar as regras de derivação: ao derivar em relação a x, tratamos y como constante, e vice-versa.

Passo 2

Para (a), calculamos as derivadas parciais gerais:
A derivada de f em relação a x é
partial fpartial x(x,y) = partialpartial xl(3x^2r) + partialpartial xl(7xyr) + partialpartial x(2x) = 6x + 7y + 2
A derivada de f em relação a y é
partial fpartial y(x,y) = partialpartial yl(-5y^2r) + partialpartial yl(7xyr) + partialpartial y(-4y) = -10y + 7x - 4

Passo 3

Para (b), basta substituir nos resultados de (a):
partial fpartial x(1,1) = 6cdot1 + 7cdot1 + 2 = 15,
partial fpartial y(-1,0) = -10cdot0 + 7cdot(-1) - 4 = -7 - 4 = -11

Passo 4

Para (c), derivamos novamente as parciais de (a):
partial^2fpartial x^2(x,y) = partialpartial x(6x + 7y + 2) = 6,
partial^2fpartial y^2(x,y) = partialpartial y(-10y + 7x -4) = -10,
partial^2fpartial xpartial y(x,y) = partialpartial y(6x + 7y + 2) = 7,
partial^2fpartial ypartial x(x,y) = partialpartial x(-10y + 7x -4) = 7

Passo 5

Para (d), avaliamos os resultados de (c) nos pontos indicados:
partial^2fpartial x^2(0,0) = 6,
partial^2fpartial y^2(0,-1) = -10,
partial^2fpartial xpartial y(1,1) = 7,
partial^2fpartial ypartial x(-1,1) = 7

Resposta

(a) partial fpartial x(x,y) = 6x + 7y + 2, partial fpartial y(x,y) = -10y + 7x - 4
(b) partial fpartial x(1,1) = 15, partial fpartial y(-1,0) = -11
(c) partial^2fpartial x^2(x,y) = 6, partial^2fpartial y^2(x,y) = -10, partial^2fpartial xpartial y(x,y) = 7, partial^2fpartial ypartial x(x,y) = 7
(d) partial^2fpartial x^2(0,0) = 6, partial^2fpartial y^2(0,-1) = -10, partial^2fpartial xpartial y(1,1) = 7, partial^2fpartial ypartial x(-1,1) = 7

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