Question

(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 a Cartesian equation for the line in R2 that passes through the points (3,0) and (0,4). (iii) Write down a Lagrangian function (2, y, 1) whose only stationary point (2*,y*, 1*) corre- sponds to the point of tangency (*,y*) between the line defined in part (ii) and the relevant contour of the objective function f. (iv) Find the coordinates (z* y*, \*) of the stationary point of your function L(x, y, 1). (v) State if the point (3*,y*) is a constrained minimum of f on D, a constrained maximum off on D, or neither a constrained minimum nor a constrained maximum of f on D. You need to justify your answer. (vi) Write down the minimum value, fmin, and the maximum value, fmax, of f on D.

Answer 1

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