parametric functions x(t) = (x1(t), x2(t))
Question 1. For each of the following parametric functions x(t) = (x1(t), x2(t)) perform the following:
a) find the tangent vector to the curve x’(t) and evaluate it at t = 1;
b) use substitution to eliminate t and write down an implicit function g(x1, x2) = 0 representing the curve and find its gradient vector, ?g(x1, x2);
c) show that the vectors x’(1) and ?g(x1(1), x2(1)) are orthogonal;
d) sketch the curve x(t) and illustrate the condition in c).
i) x(t) = (1/t, t) for t in (0, 8).
ii) x(t) = (2t, 12 – 3t) for t in [0,4].
iii) x(t) = (2 + t, 4 – t2) for t in (-8,8).
Question 2. For each of the following functions,
a) determine whether f(x1, x2) is concave, strictly concave, convex or strictly convex using the test of the definiteness of the Hessian matrix of f(x1, x2).
b) find the stationary points of f(x1, x2), and determine (if possible) whether they are maximizers or minimizers of the function.
i) f(x1, x2) = x12 + 3×22 + 5 .
ii) f(x1, x2) = 10 – x12 – x22 + x1x2 .
iii) f(x1, x2) = x12 + x22 + 2×1
Question 3. Use the method of Lagrange multipliers to find the points that solve the following problems. Check the second order conditions to ensure your proposed solution solves the problem.
i) Minimize f(x1, x2) = 2×1 + 5×2 subject to x11/2×21/2 = 20.
ii) Maximize f(x1, x2) = 2 x11/2 + x2 subject to (1/3) x1 + x2 = 100.
iii) Maximize f(x1, x2) = 6x1x2 subject to x12 + x22 = 1.