

15) Show that the fune [6] Let f : (a, b) → R be strictly convex on (a,b). Show that there is 80 cE (a, b) such that f is strictly increasing or stricty decreasing on le,b) some poirnt
15) Show that the fune [6] Let f : (a, b) → R be strictly convex on (a,b). Show that there is 80 cE (a, b) such that f is strictly increasing or stricty decreasing on le,b) some poirnt
3. Let B(p, m) ce C RTlpcS m be the budget correspondence for p >>0, m2 0. a. Show this correspondence is continuous (i.e, both uhe and Ihe) b. Show this correspondence is convex-valued and compact-valued.
3. Let B(p, m) ce C RTlpcS m be the budget correspondence for p >>0, m2 0. a. Show this correspondence is continuous (i.e, both uhe and Ihe) b. Show this correspondence is convex-valued and compact-valued.
Exercise 1. Let a, b,CE Z such that alb and alc. Show that alkb + pc for any k, p є z.
Exercise 5.5 please
Exercise 5.5: Show that the collection of all half-open intervals [a, b) where a, bER form a base for the half-open interval topology for R from Theorem 2.20. Theorem 2.20: Let H ={v I v=ø or for each xeV there is a half-open interval interval topology.
Exercise 5.5: Show that the collection of all half-open intervals [a, b) where a, bER form a base for the half-open interval topology for R from Theorem 2.20. Theorem 2.20: Let...
6. (16 points) Let CE C be a primitive n-th root of unity. Let X = 6 + 1/5. (a) (4 points) Show that Q(5) R = Q(1). (b) (4 points) Let f be the minimal polynomial of over Q. Show that Q(x) is a splitting field of f over Q. (c) (4 points) Show that Gal(Q(^)/Q) – (Z/nZ)* / (-1). (d) (4 points) Find the minimal polynomial of 2 cos(27/9) over Q.
)-( 1 (c) Let C be a real 3 x 3 matrix and b be a real 3-vector. The general solution to the matrix equation Cx=b is given by 2 2 =X3 + -4 2 for all XER Let 10 y = -6 8 (i) Let z be a real 3-vector. Find the solution set to the matrix equation Cz=0 (ii) Calculate M1, M2 ER such that 2 y = M1 ( 3 + H2 ·()--() 1 (iii) Express Cy...
Let C be the curve parameterized by with 0 ≤ t ≤ 2π.
a) Show that the curve C is contained in a plane and that it is
a closed curve. You must explicitly give the equation of the plane
that contains the curve.
*( Reminder: the general equation of a plane is ax + by + cz =
d.)
b)Let P be the plane found in a). Calculate the area of the part
of P delimited by curve C...
Please show work on how to get
the answers. Thanks in advance! :)
7. Let P(-3,1)and Q(5.6) be two points in the coordinate plane. (a) Find the distance between P and Q (b) Find the midpoint of the segment Po. (c) Find the slope of the line that contains P and Q d) Find an equation of the line in standard form that contains P and 0 (e) Find an equation of the line in standard form that is perpendicular...
7. Consider a family of maps f :R-R, where f(r)= 2+c, cE R. a) Let c 0. Find all the fixed points of f and analyze the map by drawing a cobweb. Check stability of the fixed points b) Find and classify all the fixed points of f as a function of c. c) Find the values of c at which the fixed points bifurcate, and classify those bifurcations. d) For which values of c is there an attracting cycle...
Let A E(R") be Hermitian and positive definite, let v Define g R" R by R", and let cE R (a) Show that g is polynomial function of (... ,En) and in particular it has continuous partial derivatives of all orders. (b) Show that oo. Hint: Use Ezercise Ic. (c) Prove that g(x) achieves a global minimum d) Compute Vg(x). Show that g has a unique critical point, and hence argue that the minimum must be achieved at this point....