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MATH 148, SPRING 2016 COMMON EXAM III (PART 2) - VERSION A LAST NAME: FIRST NAME: INSTRUCTOR: SECTION NUMBER: UIN: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited. 2. Present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. THE AGGIE CODE OF HONOR “An Aggie does not lie, cheat or steal, or tolerate those who do.” Signature: DO NOT WRITE BELOW! Question Points Awarded Points 1 12 2 12 3 12 4 12 Part 1 52 100 1 1. (12 pts) Use the method of Lagrange multipliers to find the maximum and minimum values of f (x, y) = x2 − 2y − y 2 subject to the constraint g(x, y) = x2 + y 2 = 1. 2 2. (12 pts) Find all local extrema and saddle points of f (x, y) = x2 + y 2 + x2 y + 4. 3 3. (12 pts) Consider the function f defined by 2x2 y f (x, y) = x4 + y 2 0 if (x, y) 6= (0, 0) if (x, y) = (0, 0) Determine whether f is continuous at (0, 0). Explain your answer. 4 4. (12 pts) Find all nonnegative equilibria of the system of difference equations x1 (t + 1) = x2 (t) x2 (t + 1) = 4x1 (t)[1 − x2 (t)] and classify each as locally stable or unstable. 5