Question

Let f (x1 , X2, X3, X4) X1 . X2-X3 . (11mP + 100x1 ex2+ รื่ 1+2)2 4-

1) Compute the gradient of f and the Hessian of f.

2) Is the Hessian positive semidefinite, positive definite, negative definite, negative semidefinite, or indefinite at the following points:

(1, 1, 5, 0) and (1, 1, 5, 2) and (1, 1, 1, 2)?

Answer 1

f(x_1, x_2, x_3, x_4) = x^4_1, x_2 - x_3 1/(1 + x_2)^2 + 100 x_1 e^x_3 + x^3_4 grad f(x_1, x_2, x_3, x_4) = (partial differential f/partial differential x_1, partial differential f/partial differential x_2, partial differential f/partial differential x_3, partial differential f/v x_4) and H f(x_1, x_2, x_3, x_4) = partial differential^2 f/partial differential x^2_1 partial differential^2 f/partial differential x_2 partial differential x_1 partial differential^2 f/v x_3 partial differential x_1 partial differential^2 f/partial differential x_4 partial differential x_1 partial differential^2 f/partial differential x_1 partial differential x_2 partial differential^2 f/partial differential x^2_2 partial differential^2 f/partial differential x_3 partial differential x_2 partial differential^2 f /partial differential x_4 partial differential x_2 partial differential^2 f/partial differential x_1 partial differential x_3 partial differential^2 f/partial differential x_2 partial differential x_3 partial differential^2 f/partial differential x^2_3 partial differential^2 f/partial differential x_1 partial differential x_3 partial differential^2 f/partial differential