Question

. Consider the function f(x, y) = 3x 2 + 7x 2 y 3 . Compute the gradient, compute the Hessian, and write down the second order approximation to this function at the point (1, 1).

Answer 1

Liny fyy 20 42 42 42 fix,y) = 3x² + fx2y3 for gradient Ex (ny) = 6x +14x43 fyrxiy) 21x²y2 gradient = (6x + 14 xy3, i + (21x242) Î at point (1,1) gradient soit 21 fu chig) = 6 +1443 +1473 Fyy (x,y) = 42x24 Faylx,y) = 42xy 2 fyx (419) = 42x²4 Hessian Matrix of Hessian of flary) Hiflry) [Fax fyn To+1443 42824 42x4² 42x²y at point (1,1) H f(x,y)= det (4 finly)] = 42 (20-42) = .42822 f (1, 1) = 10 =-924 = [-924 $40,0) = 20 fxxC!, 1) = 20 for 1) = 42 fy (1,1)= 21 fxy (1, 1) = 42, Fyy (1,1)=42 Quadratic approximation or 2nd degree aproximan - FC. V + felt, 1)(x-U +fy0, 19-1) + { xfxx(!!!) (x-v2+ fxy0,0)(x-1)09-1) = 10 + 10(x-1) + 2114-1) + 10(x-1) + 42(x-1) (-1) + 1 fyy (1, 1) (9-12 + 21 (9-12 - 1 2nd degree approximau = 10(-1)=2104-12742(x-1)(8-1) + 10(-1) +2114-1)+10 ANS