Let n be a positive integer and a, b be two real numbers. Determine (with argument) whether the following subsets of R[X] are subspaces: 1. {f(X) e R[X] | f'(8) = 0}, where f'(X) denotes the derivative of f(X). 2. {S(X) e R[X] | f(m)(0) + (S' F(X) - dX) = 0}, where f(")(X) denotes the n-th derivative of f(X). 3. {f(X) € R[X] | deg(f) = n}, where n is a fixed positive integer. 4. {f(X) E R[X] | f'(1) < f(4)}.

#2. Let flz) = (x, y) +iv(x,y) be an analytic function m a domain D. If a zeck, y) + b v(x, y) = C in D are real Constants not all 0. where a, b and c Prove that fiz) is constant in D.

Consider a function z = F(u(s, t), v(s, t)) where F, u, and u are differentiable and u(8, 6) = -1, v(8, 6) = 2, u, (8, 6) = 5, u, (8, 6) = 9, vs(8, 6) = −3, v₁(8, 6) = 4, Fu(-1,2) = -4, and F(-1,2)= -10. Compute and when s = 8 and t = 6 dt

(s) Find the linearization of f at the given point. Then estimate the value of f. (a) f(x, y) = xy² at (2, 1). Estimate f(2.001, 1.05) y? f(x, y) = at (1,2). Estimate f(0.99, 1.99)