In: Advanced Math
3. Given is the function f : Df → R with F(x1, x2, x3) = x 2 1 + 2x 2 2 + x 3 3 + x1 x3 − x2 + x2 √ x3 . (a) Determine the gradient of function F at the point x 0 = (x 0 1 , x0 2 , x0 3 ) = (8, 2, 4). (b) Determine the directional derivative of function F at the point x 0 in the direction given by vector r = (2, 1, 2)T . (c) Determine the total differential dF of function F and use it to compute approximately the absolute and relative error in the computation of F(x 0 ) when the independent variables are from the intervals x1 ∈ [7.8, 8.2], x2 ∈ [1.9, 2.1], x3 ∈ [3.9, 4.1]. (14 points) 4. (a) Determine all points satisfying the necessary conditions of the Lagrange multiplier method for a local extreme point of the function f(x, y) = x 2 + y 2 subject to the constraint x 2 + 2y 2 − 2 = 0 . (b) Using the sufficient conditions, check whether the point (x ∗ , y∗ ; λ ∗ ) = (− √ 2, 0; −1) is a local minimum or maximum point and give the corresponding function value.