Let a continuously differentiable function f: Rn → R and a point x E Rn be given. For d E Rn we define Prove the following statements: (i) If f is convex and gd has a local minimum at t-0 for every d E R", then x is a minimiser of f. (ii) In general, the statement in (i) does not hold without assuming f to be convex. Hint: For) consider the function f: R2-»R given by Let a continuously...
3. Suppose that (M, ρ) is a compact metric space and f : (M, p)-+ (M,p) is a function such that (Vz, y E M) ρ (z, y) ρ (f (x), f (y)). a. Let x E (M, ρ) and consider the sequence of points {f(n) (X)}n 1 . (Remember: fn) denotes the composition of f with itself, n times, so for each n, f+() rn, k E N) such that ρ (f(m) (x) ,f(n +k) (r)) < ε ....
Let A be n × n with AT-A. (The matrix A is syrnmetric.) Let B be 1 × n and let c E R. Define f : Rn → R by f(x) = 2.7, A . x + B . x + c. Show that The function f is a quadratic function Let A be n × n with AT-A. (The matrix A is syrnmetric.) Let B be 1 × n and let c E R. Define f : Rn...
: R → Rn be a Ci path which solves the 1. Let F : Rn → R be a C1 function, and let differential equation, E'(t)--VF(C(t)), te R. (a) Show that f(t) F((t)) is a non-increasing function of t (ie, f'(t) 30 Vt.) (b) For any t for which F(E(t)) * 0, decreasing in t (ie, f'(t) <0.) show that č is a smooth path, and f(t) is strictly
Let the function f R R be given by 1,)- f 1 z-1 Draw the graph of f versus the values of z. Is f a bijection (i.e., one-to-one and onto)? If yes then give a proof and derive a formula for f. If no then explain why not Let the function f R R be given by 1,)- f 1 z-1 Draw the graph of f versus the values of z. Is f a bijection (i.e., one-to-one and onto)?...
Question 12 11. Show that if F is continuous on Rn and F(X + Y) = F(X) + F(Y) for all X : in R", then A is linear. HINT: The rational numbers are dense in the reals. 12. Find F and JF. Then find an affine transformation G such that F(X)-G(Y) lim =0. T x2+y+2z (a) F(x, y,z)coxy. Xo- (1,-1,0) e*yz ex cos y (b) Fe*sin y 1, xo=(0, π/2) 13. Find F. g1 (x) 11. Show that if...
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
-n ', S Let f(x,yZFz2_xy. Let v=<1,1,1>. Let point P=<2,1,3> a. Compute gradient of fx,y,z) b. If the contours are far apart, is the length of the gradient large or small? Answer: Explain! What MATLAB command is used to draw the gradient vectors? Answer: - c. Compute the directional derivative in the direction of v. d. Compute the equation of the tangent plane to f(x,y,z) at the point P. e. Use the chain rule to compute r if x t2,...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
3. In this problem we consider only functions defined on the real numbers R A function f is close to a function g if r e Rs.t. Vy E R, A function f visits a function g when Vr E R, 3y E R s.t. For a given function f and n E N, let us denote by fn the following function: Below are three claims. Which ones are true and which ones are false? If a claim is true,...