Question

9. For a finction -x determine if the function is globally concave or convex.

Answer 1

Let f be a function of two variables with continuous partial derivatives and cross partial derivatives on set S and denote the Hessian of f(x1,x2) at the point (x1,x2) by H(x1, x2) . Then

f is concave if and only if H(x1,x2) is negative semidefinite for all (x1,x2)∈ S

if H(x1,x2) is negative definite for all (x1,x2) ∈ S then f is strictly concave

f is convex if and only if H(x1, x2) is positive semidefinite for all (x1, x2)∈ S

if H(x1, x2) is positive definite for all (x1,x2)∈ S then f is strictly convex.

Let f be a function of two variables with continuous partial derivatives and cross partial derivatives on set S and denote the Hessian of f(x1, x2) at the point (x1, x2) by H(x1, x2) . Then f is concave if and only if H(x1, x2) is negative semi definite for all (x1, x2) element of S if H(x1, x2) is negative definite for all (x1, x2) ? S then f is strictly concave f is convex if and only if H(x1, x2) is positive semi definite for all (x1, x2) element of S if H(x1, x2) is positive definite for all (x1, x2) element of S then f is strictly convex. F(x_1, x - 2) = ln(e^3 ln x_1 e^5 ln x_2 (e^0 - e^ln x_1 - x - 2)^6) = ln (x_1^3 x_2^5(1 - x_1 - x_2)^6) = ln x_1^3 + ln x_2^5 + ln(1 - x_1 - x_2)^6 f(x_1, x_2) = 3 ln x_1 + 5