Investigate the following func Mon por Local maximun and Local minimum point. 3) RCX, Y, 2 ) = x² + y² + 2² + xy +xz + y2-x-Y-2

Investigate the pollowin punction for Local Maximum and Local minimum points. 2 2 2) P² (X, Y,₁ ²) = -x² - y²-z² + 2x +2y-22

4. Use the following steps to find the dimensions of the rectangular closed box with maximal volumes that is made with 36 square feet of material. In your answer please label the steps and box your answer for each step so I can allocate points easily. (a) Write the volume and surface area of the box as a functions of three variables (label volumes as V(x, y, z) and surface area as S(x, y, z). (b) Use the constraint of the problem on S to rewrite the volume of the box as a function of just x and y. (c) Now, we can disregard any pairs (x, y) that have either coordinate as a negative number or zero. Explain why this is. (d) Find the critical points of V(x, y) with x > 0 and y > 0. (e) What are the dimensions of the box with maximal volume?