7. Define a function G: R XRRXR as follows: G(x,y) = (2y, -x) for all (x,y)...
2. Define a function g: R3 +R by g(x, y, z) = 2x2 + y2 + x2 + 2xz – 2y – 4. (a) Find all the critical points of g. (b) Compute the Hessian H, of g. (c) Classify the critical points of g. (d) The surface g(x, y, z) = 0 is an ellipsoid . Use the method of Lagrange multipliers to find the maximum value of the function (5 marks) (5 marks) (5 marks) f(x, y, z)...
4) Define the function f :Rx R + RxR by f(x, y) = (x + y, x – 2y). Prove that f is a bijection.
Define a relation R from R to R as follows: For all (x, y) E R x R, (x, y) E R if, and only if, x= y2 + 1. (a) Is (2, 5) E R? Is (5, 2) e R? Is (-3) R 10? Is 10 R (-3)? (b) Draw the graph of R in the Cartesian plane. (C) Is R a function from R to R? Explain.
Define the
function f : Rf3 ! Rf5 by
f(x)= 5x/x-3Prove: f is surjective ("onto"
R\5).
R {5} by 7. (15 pts) Define the function f : R\{3} f(x) = 0 Prove: f is surjective ("onto" R\{5}). I
8. Let V = {(x,y)x,y e R}. Define addition on V as follows: (x,x)+(x2,)=(x, +x,-1,, +y,+3) [4 marks] a. Prove addition axiom #3 (Addition is commutative). b. Find the zero vector.
how do u do 6?
F-'(C-D)= F-'(C)-F-'(D). 4. (10 points) In following questions a function f is defined on a set of real numbers. Determine whether or not f is one-to-one and justify your answers. (a) f(x) = **!, for all real numbers x #0 (6) f(x) = x, for all real numbers x (c) f(x) = 3x=!, for all real numbers x 70 (d) f(x) = **, for all real numbers x 1 (e) f(x) = for all real...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
Find a matrix A that completely determines the function T(x, y) = (2y − 3x, x − 4y, 0, x). Determine if T is one-to-one and onto.
6. Plot the 3D surface and contour levels of the following function: z(x, y)cos(2y-x) sin(2x) such that-π x π and-r y < π [10 marks] 7. Create a 5 x 5 random matrix M6 with elements ranging from 10 to 33. Using indexing, define the following arrays: An array containing all elements of M6 that are greater than 3 and smaller 6 marks] An array containing all elements of M6 that are negative or between 29 and 33. 6 marks...
Problem 5. Define a relation ~on R x R as (x, y) ~(a,b) if and only if either x-a or y- b. Prove or disproof, isan equivalence relation? If so, write down all the equivalence classes.