Jacobian Matrix - Hessian Matrix - System of Equation - Implicit Function Theorem - Mathematics Assignment Help

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Mathematics Assignment Help

Task: Problem 1 1. Find the Jacobian matrix of F in the following cases.     2. Find the Hessian matrix of f in the following cases       Problem 2 Consider the system of equations     1. Show that the z and t are determined as a function of x and y near the point (0, 1, 1,-1). Can we apply the Implicit Function theorem? 2. Compute the partial derivatives of z and t with respect to x, y at (0, 1). 3. Without solving the system, what is the approximate value of z(0.001, 1.002) (Hint: Use the first order Taylor approximation about the point (1,0) to find the approximation) 4. Compute ? 2z/?x?y (0, 1) Problem 3 The income function is R(x, y) = x(100 ? 6x) + y(192 ? 4y) where x and y are the number of articles sold. If the cost function is C(x, y) = 2x 2 + 2y 2 + 4xy ? 8x + 20 determine the maximum profit. Problem 4 Minimize x 4 + y 4 + z 4 on the plane x + y + z = 1 Problem 5 Solve the optimization problem M in x2 + y 2 ? 20x s.t. 25x 2 + 4y 2 ? 100

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