[Risolto] Hessiano

A) f(x, y) = z = x^2 - y^3 + 3*x*y
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A1) gradiente
nabla[f(x, y)] = {2*x + 3*y, 3*(x - y^2)}
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A2) punti critici
nabla[f(x, y)] = 0 ≡
≡ {2*x + 3*y, 3*(x - y^2)} = {0, 0} ≡
≡ O(0, 0) oppure C(9/4, - 3/2)
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A3) Hessiana
nabla[2*x + 3*y] = {2, 3}
nabla[3*(x - y^2)] = {3, - 6*y}
H = {{2, 3}, {3, - 6*y}}
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A4) hessiano
h(x, y) = det[H] = - 3*(4*y + 3)
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A5) test delle derivate seconde
h(0, 0) = - 3*(4*0 + 3) = - 9 < 0 ≡ autovalori discordi ≡ punto di sella
h(9/4, - 3/2) = - 3*(4*(- 3/2) + 3) = 9 > 0 & H[1, 1] > 0 ≡ punto di minimo
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A6) verifica
http://www.wolframalpha.com/input/?i=extrema+x%5E2-y%5E3%2B3*x*y
http://www.wolframalpha.com/input/?i=saddle+points+x%5E2-y%5E3%2B3*x*y
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B) (z = y + x^2 + y^2 - 4*x) & (x + 1 - y = 0) ≡
≡ (y = x + 1) & (z = y + x^2 + y^2 - 4*x) ≡
≡ z = 2*x^2 - x + 2
* z' = 4*x - 1
* z'' = 4 > 0
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B1) test delle derivate seconde
* (z' = 0) & (y = x + 1) & (z'' > 0) ≡
≡ (x = 1/4) & (y = 5/4) & (z'' > 0) ≡ punto di minimo
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B2) verifica
http://www.wolframalpha.com/input/?i=extrema+y%2Bx%5E2%2By%5E2-4*x+where+x%2B1-y%3D0