Homework SourseConsider the problem of mazimizing the function fxy 2x 3y
Consider the problem of mazimizing the function f(x,y) = 2x + 3y subject to the constraint ?x + ?y = 5.
a)Try using the lagrange multipliers to solve the problem.
b)Does f(25,0) give a larger value than the one in part (a)?
c)Solve the problem by graphing the constraint equation and several level curves of f.
d)Explain why the method of Lagrange multipliers fails to solve the problem.
e)What is the significance of f(9,4)?
a)Try using the lagrange multipliers to solve the problem.
b)Does f(25,0) give a larger value than the one in part (a)?
c)Solve the problem by graphing the constraint equation and several level curves of f.
d)Explain why the method of Lagrange multipliers fails to solve the problem.
e)What is the significance of f(9,4)?
Solution
?(x,y,?) = f(x,y) + ?·(g(x,y) - c) ?(x,y,?) = 2x + 3y + ?·(vx + vy - 5) 1) ??/?x = 2 + ?/[2vx] = 0 2) ??/?y = 3 + ?/[2vy] = 0 3) ??/?? = vx + vy - 5 = 0 By 1: ? = -4vx By 2: ? = -6vy Therefore: 2vx = 3vy 4x = 9y y = (4/9)x By 3: vx + (2/3)vx = 5 (5/3)vx = 5 vx = 3 x = 9 y = (4/9)*9 = 4 Local Max: (9,4)
