Each problem is worth 3 points.

1. Find the highest and lowest points on f(x,y) = x²+y² subject to the constraint z=2x+2y.
 

2. [Stewart 12.8 #2] Find the maximum and minimum values of f(x,y) = 2x+y subject to the constraint x²+4y²=1.
 

3. [Stewart 12.8 #4] Find the maximum and minimum values of f(x,y) = x²+y² subject to the constraint x⁴+y⁴=1.
 

4. [Stewart 12.8 #6] Find the maximum and minimum values of f(x,y,z) = x-y+3z subject to the constraint x²+y²+4z²=4.
 

5. [Based on Stewart 12.8 #18] (a) Use a graphing calculator or computer to graph the circle x²+y²=1. On the same screen graph several curves of the form x²+y=c (for various values of c) until you find three that just touch the circle. What is the significance of the values of c for these three curves?

(b) Use Lagrange multipliers to find the extreme values of f(x,y)=x²+y subject to the constraint x²+y²=1. Compare your answers to those in part (a).