3. Given a second-order ODE, [x1,x2] ER?, study the stability of the equilibrium point at the...
Prove that each of the following sets is convex (a) {(x1, 22, x3) E R3 | 0 < 띠, x2, 23 and x1 + 2x2 + 3x3 6)
Suppose investment 1 (represented by X1) cannot be selected unless both investments 2 (represented by X2) and 3 (represented by X3) are also selected. If investments 2 and 3 are selected, investment 1 does not have to be selected. Which constraint would enforce this condition? O 2X1 <-X2+X3 О x1--2x2 + 2x3
3. Consider the following LP. Maximize u = 4x1 + 2x2 subject to X1 + 2x2 < 12, 2x1 + x2 = 12, X1, X2 > 0. (a) Use simplex tableaux to find all maximal solutions. (b) Draw the feasible region and describe the set of all maximal solutions geometrically.
The joint density of random variables X1, X2 is given by fx1,x2 (x1, 2)= 6x1, for 0 < xı < 1, 0 2 <1 - r Let Y X1X2. Find the joint density of Yi and Y2 Х1, Y?
Solve the following using graphing techniques: a. Maximize 2x1 + 3x2 subject to the constraints, 2x1 + 2x2 < 8,X1 + 2x25 4, and X1 > 3, x2 > 0
5.9 Draw an x1,x2 coordinate system and show the convex region which satisfies the constraints that follow: X2 20 and 0 < x < 3 -- X1 + X2 51 X1 + X2 54 Of the points in this region, find the point (or set of points) which yields the maximum of: a. P= 2x1 + x2 b. P = x1 + x2 C. P = xi + 2x2
3. Let (X1, X2) have the joint p.d.f 1 if 0 <1,0 < <1 f(1, ) else Find P(X1X2 < 0.5)
please answer the question by computer Problem 2. (25 points) Consider the following integer nonlinear programming problem. max 2 = x1x2x s.t. X1 + 2x2 + 3x3 < 10, X1 >1, x2 > 1, X3 >1, X1, X2, X3 are integers. Use dynamic programming to solve this problem.
Let X = (X1, X2) be a bivariate normal random vector such that Mi = 4,42 = 6,01 = 25, 02 = 16 and p= 0.7. 1. Find P(X2 <5|X1 = 3).
3. Use the two-phase simplex method to solve the following LP. Min z = x1 + 2x2 Subject to 3x1 + 4x2 < 12 2x1 - x2 2 2 X1, X2 20