Pls answer this with the equations and the steps. Thank you.

1. (5) Show that each of the following functions is a solution of the wave equation utt = Cuxx. (a) u = sin (kx) sin(ckt) (b) u = sin(x - ct) + In(x + ct) 2. (5) Let P = Vu2 + v2 + w2, u = xe", v = yet, and w = ery. Use the Chain Rule to find OP - and - OP ax ay when x = 0, y = 2. 3. (5) Use the Implicit Differentiation to evaluate Oz/Ox and Oz/dy, given that yz + xlny = 22. 4. (5) Find the rate of change of f(x, y, z) = y?exyz at the point P = (0, 1, -1) in the direction of u = (13, 13: 13). 5. (5) Compute the directional derivative of g(r, s) = tan '(rs) at (r, s) = (1, 2) in the direction of v = 52 + 10j. 6. (5) Find the local maximum and minimum values and saddle point (s) of the function f (x, y) = y cos x. 7. (10) Find the absolute maximum and minimum values of f(x, y) = x ty- xy on the closed triangular region with vertices (0, 0), (0, 2), and (4, 0). 8. (5) Let f(x, y, z) = x2 + y2 + 22. Use Lagrange multipliers to find the maximum and minimum values of f subject to the constraint a ty + z = 12. 9. (10) Find the extreme values of f(x, y) = 2x2 + 3y2 - 4x - 5 on the closed disk D = {(x, y) ER2 | 22 + 2