2 (b) Prove that + 3 cos(atx) O has at least two solutions with x €...
At least one of the answers above is NOT correct. (1 point) Suppose /(x) = x + 3x + 1. In this problem, we will show that has exactly one root (or zero) in the interval (-3,-1). (a) First, we show that f has a root in the interval (-3,-1). Since is a continuous function on the interval (-3, -1) and f(-3) = and f(-1) = -1 the graph of y = f(x) must cross the X-axis at some point...
iu [5 marks]b ii)) Consider the function, f, as the followings,vRuǐ (x1, x2)-5xỈ + x-. + 4x1 x2-14x1-6x2 + 20 ( !0% x») This function has its optimal solution atx"= (1,1) and f(1, 1) 10. Run the k-th iterates of the Newton algorithm, and compute the descend the k-th iteration (dk). [5 marks] Resource Allocation prob iu [5 marks]b ii)) Consider the function, f, as the followings,vRuǐ (x1, x2)-5xỈ + x-. + 4x1 x2-14x1-6x2 + 20 ( !0% x») This...
Assignment 6 1. Prove by contradiction that: there are no integers a and b for which 18a+6b = 1. 2. Prove by contradiction that: if a,b ∈ Z, then a2 −4b ≠ 2 3. Prove by contrapositive that: If x and y are two integers whose product is even, then at least one of the two must be even. Make sure that you clearly state the contrapositive of the above statement at the beginning of your proof. 4. Prove that...
(a) (4 marks) Consider the function S(x) = x-cos(x). 1) Prove that S has at least one zero in the interval [0, ) f(0)f(x) <0. (O)f(x) > 0 and is continuous f(0)f(x) < 0 and is continuous. ii) Prove that S has at most one zero in the interval [0, x) f' <0 on (0,#] so that is strictly increasing on (0,r)- 1'>0 on (0,#] so that f is strictly increasing on (0, #) 1'>on (0,r) so that is strictly...
1. Use Pigeon hole principle to prove that any graph with at least 2 vertices contains two vertices of the same degree. (Hint: Prove by contradiction. (4 points) 2. Given (6 Points) a. Prove the above equation using binomial theorem. (3 Points) b. Give a combinatorial proof for the given equating. (3 Points) 4n = (0)2" + (1)2" +...+)2"-
8. Let p be an odd prime. In this exercise, we prove a famous result that characterizes precisely when -1 has a sqare root 1 mod 4. (You will need Wilson's Theorem for one (mod p). Prove: a 2--1 mod p has a solution if and only if p dircction of the proof.) 8. Let p be an odd prime. In this exercise, we prove a famous result that characterizes precisely when -1 has a sqare root 1 mod 4....
Problem 2 (20 points). Prove that a polynomial of odd degree has at least one real root. (Hint: Use Intermediate Value Theorem.)
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
Find the solutions for cos(2?)=3−sin2(?)−5cos(?)−cos2(?)cos(2x)=3−sin2(x)−5cos(x)−cos2(x), in the interval [0,2?).[0,2π). The answer(s) is/are ?= 5.5 Solutions of Trig Equations: Problem 17 Previous Problem Problem List Next Problem (1 point) Find the solutions for cos(2x) = 3 – sin?(x) - 5 cos(x) - cos(x), in the interval [0, 21). The answer(s) is/are x = Note: If there is more than one solution enter them separated by commas. If needed enter a as pi.
For the equation 3 - 2x = ex - cos(x) 1. Use the intermediate value theorem to show the equation has at least one solution 2. Use the mean value theorem to show that the equation has at most one solution