1. Consider f (x, y, z) = xy + xz subject to the constraint x- + y- + 2- = 32. Find the absolute minimum and absolute maximum value of f. 2. Minimize f(r, h) = 2mr? + 2mrh subject to the constraint r=h = 54, wherer > 0 and h > 0. 3. A company manufactures two types of bicycles, a racing bicycle and a mountain bicycle. The total revenue from x, units of racing bicycles and xz units of mountain bicycles is R = -6x13 -10x2 - 2x1 X2 +32x1 +84x2 where x, and x2 are in thousands of units. Find x, and x2 to maximize the revenue. 4. One way to measure species diversity is to use the Shannon diversity index H. If a habitat consists of three species, A, B, and C, then its Shannon diversity index is H = -xinx - ylny - zinz where x is the percent of species A in the habitat, y is the percent of species B in the habitat, and z is the percent of species C in the habitat. Use the fact that x + y + 2 = 1 to show that the maximum value of H occurs when x = y = 2 = $. What is the maximum value of H? 5. Find the limit (if it exists) and discuss the continuity of the function. lim (xy)-(11) x +yi 6. Find all first partial derivatives of f (x, y) = vx - y? - zz