Q1.

2 5 For the function f ( x, , x 2 , x 3 ) = X, Xz X3 determine the gradient vector . Then calculate Second order partial derivatives and arrange in matrix notation.For the function f (x1, X2, X3) = aix tazx7 - 2 + a3x3, determine the first- and second-order partial derivatives, and arrange in vector/matrix notation.Use theorem 11.5 to show that the function f(x1, X2) = 10 - x2 - x2 is strictly concave (see example 11.27). Example 11.27 Use the sign of the second-order total differential to show that the function y = 5 - (x1 + x2)2 is concave. Solution First- and second-order partial derivatives of this function are fi = -2(x1 + x2), f2 = -2(x1 + x2) fil = -2, f22 = -2, f12 = -2 It follows that 1=a-x day = fudx, + 2findx, dx2 + f2 dx2 *2 = -2dx - 4dx, dx2 - 2dx3 = -2(dx1 + dx2)2 0 and so the function is concave. To see that this function is not strictly concave, Figure 11.20 Linear segment on notice that along the set of (x1, x2) values satisfying x2 = a - x1, where a is any the surface of the graph of y = 5 - (x] + x2)" in example 11.27 constant, we have y = (x, ta - x1)' =a . These (x1, x2) values generate hori- zontal linear segments of the graph of f as illustrated in figure 11.20. Theorem 11.5 If the function y = f(x1, x2) defined on R2 is twice continuously differentiable and d'y = fudx; + 2f12 dx] dx2 + f2z dx;