Question

(b) f(x, y) = 2.2-2y2-6x + 8y + 3 (c) f(z, y)=x2 + 6ry + 2/-6x +10y-2 Constrained Optimization 3. Find the values of r and y that maximize (or minimize) f(x,y) subject to the following constraints (a) f(z, y) = ry, subject to x + 2y = 2 (b) f(z, y)y+4), subject to z+y 8 (c) f(x,y) = x-3y-xy, subject to x + y = 6 (d) f(x, y)-7-y 2, subject to z ty 0 (e) f(z, y- x2 + y2 = 4 -+y, subject to Applications the farmer's market. The farmer's market only o competition. She currently sells lemonade for $1. 4. Jane is running a lemonade stand at allows one stand, so she faces n Tarzan tells her that this price is too high and suggests that if she lowers the price she

Questions A and E

Answer 1

a)

e) f(x, y) = x + y^2 st x^2 + y^2 = 4 f = x + y^2 + lambda (x^2 + y^2 - 4) F_x = 1 + lambda (2 x) = 0 2 lambda x = -1 2 x = -1/lambda lambda = -1/2 x rightarrow (1) F_y = 2 y + lambda (2 y) = 0 2 y - 2 lambda y = 0 2 y(1 - lambda) = 0 2 x + 1/2 x = 0 2 x = -1 x = -1/2 putting the value in constraints (- 1/2)^2 + y^2 = 4 (1/4) + y^2 = 4 y^2 = 4 - 1/4 y^2 = 15/4. y = squareroot 15/2.

e)