1) Find and classify the critical points of each function. a) f(x, y) = x²+xy + y²+y b) g(x,y) = re-2(x²+y2 c) h(x,y) = 3-12xy + 8y³ d) r(x,y) y cos e) s(x, y) = sin x sin y 2) Determine the absolute maximum and absolute minimum of the function over the region R. in R². a) f(x, y) = x²+ y²- 2x, with R a closed triangular region with vertices (2,0), (0, 2), and (0,-2). b) g(x,y) = x²+ y²+2y+4, with R= [−1,1] × [−1,1]. c) h(x,y) = r²y, with R = {(x,y): ≥0,y≥0, and r²+y2 ≤3} d) r(x,y) = 2r+y, with R= {(x,y): x² + y² ≤ 1} 3) Determine the maximum and minimum values of the function subject to the given constraint. a) f(x, y) = x²+ y², with xy = 1. b) g(x,y)=3x+y, with x²+ y² = 10. c) h(x,y) y² ², with r²+4y² = 4. d) q(x,y), with a² + y² = 16. e) r(x, y, z) =2y22, with r² + y²+2=1. 1 4) a) Find the shortest distance from the plane described by x+y+z = 1 to the point (2,0,-3). b) Find the points on the sphere described by x²+ y²+22=4 that are closest to and farthest from (3,1,-1). c) Find the dimensions of a rectangular prism with a volume of 1000 that has the minimum surface area. d) Find the dimensions of a rectangular prism with a surface area of 64 that has the maximum volume. 5) a) Show that the rectangle with fixed perimeter P that has the largest area is a square. b) Show that the triangle with fixed perimeter P that has the largest area is equilateral.