Math 121a midterm 1 by using lagrange multipliers find the

Question

Math 121A: Midterm 1-

1. The function f(x, y, z) = x² + 3y² + 5z² + 2xy + 4yz - 4x - 2z, has one minimum point. Find its location.

2. (a) Calculate the derivative of f(x) = log(log(log(x))).

(b) By using an appropriate series test, determine whether

_n=3Σ^∞(1/n log(n)log(log(n)))

converges or diverges.

(c) By using an appropriate series test, determine whether

_n=3Σ^∞((-1)ⁿ/n log(n)log(log(n)))

converges or diverges.

3. By using Lagrange multipliers, find the smallest possible surface area (including both ends) of a cylinder with volume V.

4. (a) By considering appropriate powers of e^iθ = cosθ + i sinθ or otherwise, determine an expression for sin³θ as a linear combination of terms with the form sin kθ.

(b) Consider the annulus A defined as a ≤ r ≤ b in polar coordinates, where 0 < a < b. Show that for any integer k, the function r^±ksin kθ is a solution to the Laplace equation ∇²φ = 0 in A.

(c) Find a solution to ∇²φ = 0 in A that satisfies the boundary conditions

φ(a, θ) = 4 sin³θ, φ(b, θ) = 0.