Consider the minimisation and maximisation of the objective function f : R2 + R given by...
Consider the minimisation and maximisation of the objective function f : R2 + R given by f(x,y) = (1 - 1)2 + y2 + 3 on the feasible region D C R2 consisting of the boundary and interior of the right-angled trian- gle whose vertices are the points (0,0), (3,0) and (0,4). (a) Write down a Lagrangian function L(x, y, 1) whose only stationary point (x*, y*, \*) corre- sponds to the point of tangency (r*, y*) between the line...
(a) Consider the minimisation and maximisation of the objective function f : R2 + R given by f(x,y) = (x - 1)2 + y2 +3 on the feasible region D C R2 consisting of the boundary and interior of the right-angled trian- gle whose vertices are the points (0,0), (3,0) and (0,4). (0) Make a sketch on the coordinate plane Rd of the region D and add to your sketch a few contours of the objective function f. (ii) Obtain...
Find the extreme values of the function f(x, y) = 3x + 6y subject to the constraint g(x, y) = x2 + y2 - 5 = 0. (If an answer does not exist, maximum minimum + -/2 points RogaCalcET3 14.8.006. Find the minimum and maximum values of the function subject to the given constraint. (If an answer does not exist, enter DNE.) f(x, y) = 9x2 + 4y2, xy = 4 fmin = Fmax = +-12 points RogaCalcET3 14.8.010. Find...
Find the maximum and minimum of the objective function: F =3x+2y subject to constraints: x > 0 y > 0 x + 2y < 4 x - y<1 Maximum value = 8, at point (0,4) Minimum value =0, at point (0, 0) Maximum value = 8, at point (8/3, 0) Minimum value =0, at point (1, -3/2) Maximum value = 8, at point (2, 1) Minimum value =0, at point (-2/3, 1) Maximum value = 8, at point (2, 1)...
Question 1 (10 points). Determine the absolute minimum and maximum values of the function f(x, y) = 2x2 – 2xy + y2 – 2y +7 on the closed triangular region with vertices (0,0), (3,0), and (0,3). Be sure to show all calculations.
2. Consider the function f : R2 → R2 given by. (x,y) (a) Compute the Df(x, y) (b) List every vector r e R2 such that Df(ri, r2) 0. What can we say about the tangent plane to the surface of the graph at (ri,2,f(r1, r2))? (c) How do you know that the Hessian, Df(x, y) is necessarily symmetric? Recall that t,y D2 f(x,y) , y) (d) What are the eigenva of D2f(r1,r2) for each root of the gradient that...
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y) = if (x, y) (0, 0) (a) Prove that f is continuous at (0,0) (b) Calculate the partial derivatives (0,0) and (0,0) directly from the definition of partial derivatives. (c) Prove that f is not differentiable at (0,0).
Q1: Consider the minimisation of the following function of two variables: f(t, z.) %3D — In(1+ 7) — Т2. Subject to the linear constraints: 2я1 + х2 < 3;B х, 22 2 0. (a) Prove that this is a convex minimisation problem (b) Write down the Karush-Kuhn-Tucker conditions for this problem. (c) Find all solutions of the above KKT conditions (d) Are the solutions you found a local or a global minimum (maximum)? Justify your answer. Q1: Consider the minimisation...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
Consider the domain S ⊂ R2, determined by the following system of inequalities: x + 5y ≤ 5 ,2x + y ≤ 4 ,x + y ≤ 15 ,x ≥ 0, y ≥ 0 a) Sketch the domain S b) Find the coordinates of all “corners” (vertices of the boundary) of S c) Determine the maximum value on S of the function z = f(x,y) = 3x + 5y. If you think that a maximum value does not exist, explain...