Lagrange multiplier method A manager is planning to spend 10

(Lagrange multiplier method) A manager is planning to spend $10,000 on advertising. It costs $3,000 per minute to advertise on TV and $1000 per minute to advertise on radio. If he buys x minutes of TV advertising and y minutes of radio advertising, then the potential revenue in thousands of dollars is given by: f(x, y) = 8x + 3y + xy - 2x^2 - y^2. Using Lagrange multiplier technique, find optimal values of x and y that maximize revenue, f(x.y). and also find the maximum revenue. Suppose the cost of TV advertising increases to $3200/minute. Using sensitivity analysis, find an approximate maximum revenue.

Solution

a) r = df/dx = 4 + y - 4x

t = df/dy = 3 +x -2y

s= df/dx.dy = -6

rt - s^{2 >0}

x = 6 y= 8