Exercise 3.5 Let P = {x € R³ | x₁ + x₂ + x3 = 1, x ≥ 0} and consider the vector x = (0, 0, 1). Find the set of feasible directions at x.
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A: So, please refer this solution as it is complete solution.
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Q: Assignment: Find all exact solutions to the following equations: 2e ³(3x+5) cos(x³ + 1) = 0
A: To Find:Exact solution of the above equation:
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A: Solution:
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A: Function :
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A: No such curve exists.This is from the famous JORDAN CURVE THEOREM.Jordan curve theorem: It states…
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A: As you have mentioned I am solving #11
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Q: 1) Find the general solution of the equation Uzy + Uz Jesent 2€²x+y
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A: We simply use the defination of Laplace transform of a function.
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- Prove that if x0 is a critical point of f and Hf (x0) is positive de nida, then f has a local minimum at x0. Where Hf is the Hessian matrixLet A(t) be a n × n matrix whose coefficients are continuous functions of t on the interval (α, β). Let t0 ∈ (α, β) and asssume that y(t) is a solution of the initial value problem x' = A(t)x, x(t0) = 0. Show that y(t) = 0 for all t ∈ (α, β).What conditions on the eigenvalues of an n×n matrix A would guarantee that the system x′ = Ax has at least one solution satisfying x(t) = x0 for all t, where x0 is a constant vector?
- The function f (x, y) = 2xy certainly has a saddle point and not a minimum at (0, 0). What symmetric matrix S produces this f? What are its eigenvalues?Given two m ×m matrix X and Y , where XY = Y X.1) Let u be an eigenvector of X. Show that either Y u is an eigenvector of X orY u is a zero vector.Apply the alternative form of the Gram-Schmidt orthonormalization process to find an orthonormal basis for the solution space of the homogeneous linear system. x1 + x2 − 2x3 − 2x4 = 0 2x1 + x2 − 4x3 − 4x4 = 0 U1 = ? U2 = ?
- Find the minimum and maximum values of Q(x) under the constraint ||x||=1 and find the eigenvectors that these values occur at.Let f(x) = xT Ax be a quadratic form with associated n X n symmetric matrix A. Let the eigenvalues of A be λ1 >= λ2>= ···>= λn Then the following are true, subject to the constraint II x|| = 1: Prove The minimum value of f(x) is λn, and it occurs when x is a unit eigenvector corresponding to λnProve that the eigenfunctions of the Sturm-Liouville Problem (p(x)y')' + q(x)y + As(x)y = 0, a ≤x≤ b, with the boundary conditions y(a) = y(b) = 0 are orthogonal.
- Let w(x) = 1 and α = β = γ = 1. Find the nontrivial stationary paths, stating clearly the eigenfunctions y (normalised so that C[y] = 1) and the values of the associated Lagrange multiplier.What is the eigenvalue of the operator T, given by T(x) = 3x + 2, for the vector x = [1, 2]?Given two m ×m matrix X and Y , where XY = Y X.1) Let u be an eigenvector of X. Show that either Y u is an eigenvector of X orY u is a zero vector. 2) Suppose Y is invertible and Y u is an eigenvector of X. Show u is an eigen-vector of X.