Q23-25 with full solution thx

12:57 learn-ap-southeast-1-prod-fleet02-xythos.s3.ap-southe - Privat 5 (b) Suppose f is differentiable at x = 0. Find a. 22. (10 points) Suppose the equation xy = Inx + y3 implicitly determines a function y = y(x) near the point (1, 0). (a) Find - at (1, 0). (b) Find day da-2 at (1, 0). 23. (10 points) Solve the following problems separately. (a) Let f(x) = x3 - 3x2 - 9x + 1. Find all the critical point(s) of f, and use first derivative test to determine whether each critical point is a local minimum, maximum, or neither. (b) Let g(x) = (x - 3)er + e2. Find all the critical point(s) of g, and use second derivative test to determine whether each critical point is a local minimum, maximum, or neither. 24. (10 points) Let R be the region in ry-plane bounded by y = 2* - 1, x = 1 and T-axis. (a) Find the area of R. (b) Express the volumes of the following solids as definite integrals (Do not eval- uate): (i) The solid obtained by revolving R about the r-axis. (ii) The solid obtained by revolving R about the y-axis. 25. (10 points) (a) Find the Taylor series generated by the function 1 - 3 x - at x = 0, and express your answer in summation notation. (b) By using (a), find the Taylor series generated by the function (1 - 3x)2 at x = 0, and express your answer in summation notation. Find the radius of convergence of the series obtained. 26. (14 points) (a) Show that arctan x - arctany y 2 0. 6 (b) Use the result in part (a) to show that I