EXERCISE 5. Consider the function f(x,y) = x(1- y²) in the domain {(x, y): r² + y² = 1}. (a) Indicate if you have reasons to assess existence or nonexistence of mar and min; (b) compute mar and min (if they exist); (c) estimate mar and min (those which exist) when the constraint be- comes x² + y² = 1.4.

LT. Find the maximum value of Q(x) subject to the constraint x'x = 1. Q(x) = 14x; + 14x, + 18x, + 26x, x2 + 18x, xg + 18x2X3 О А. 25 О В. 9 О с. 1 O D. 36

Find the maximum and minimum values of the function f(x, y, z, t) = x + y + z +t subject to the constraint x2 + y? + z² + t² = 225. Maximum value is | , occuring at points (positive integer or "infinitely many"). Minimum value is occuring at points (positive integer or "infinitely many").

Let f(x.y) be a two variable function with fx = y and fy =x. What are the extreme points of f with constraint 4x2 + y² = 8? O a. (1,-2), (-1,-2), (-2, 2), (-1,-1) O b.(1,-2). (1,2). (-1, 2). (-1,-2) O C. (1,2), (1, -2). (2,1), (2,-1) O d. (1,1), (1,-1), (2,2), (2.-2)

Find the minimum value of x +y +z² givenx+y+z=3a.,

let (x) be a convex function. An optimization problem is given by min x€S subject to gi (x) >= 0, i=1,2,...,m. Show that it is a convex optimization problem if all the constraint functions gi are concave

Compute the abscissa of the minimum point of the curve  y = x3 - 12x -9.

Determine the equation of the quartic function with the following characteristics x-intercepts at x = 1, 10 a double intercept at x = -2 a y-intercept at y = 1

A certain country uses a progressive tax system. The amount of tax consists of a linear part proportional to the income and a nonlinear part depending on the income by a power law. The total amount of tax is determined by the formula T(W)=aW+(bW+c)p, where W is the income; p is the exponent, a,b,c are some positive numbers. At what level of income the tax rate will be minimal?

Max. Z= 12X, + 14X2 Subject to: 2X, + 3X2 s 24 (Constraint of raw material R1) 2X1 + X2s 16 (Constraint.of raw material R2) X1s 7 (Demand for product X1) X1 X2 2 0 1- By using the Algebraic Analysis, find whether a change in the right- hand side for constraint of raw material R2, (16), and the current basis remains optimal. 2- Change this linear programming model to Dual Problem Model, and find the values of the variables and W by using the optimal simplex tableau which shown below. B.V. X X2 S1 S3 X1 -1/4 3/4 6. X2 1/2 -1/2 4. S3 1/4 -3/4 11 1.