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 gradientof 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 thepoint 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 tocompute approximately the absolute and relative error in thecomputation of F(x 0 ) when the independent variables are from theintervals x1 ? [7.8, 8.2], x2 ? [1.9, 2.1], x3 ? [3.9, 4.1]. (14points) 4. (a) Determine all points satisfying the necessaryconditions of the Lagrange multiplier method for a local extremepoint of the function f(x, y) = x 2 + y 2 subject to the constraintx 2 + 2y 2 ? 2 = 0 . (b) Using the sufficient conditions, checkwhether the point (x ? , y? ; ? ? ) = (? ? 2, 0; ?1) is a localminimum or maximum point and give the corresponding functionvalue.
