(a) Show that g maps the interval [1, 2] into the same interval [1, 2], which...

Transcribed Image Text:

(a) Show that g maps the interval [1, 2] into the same interval [1, 2], which means to show that 1 ≤ g(x) ≤ 2, for all x € [1,2]. (b) Find a theorem, or part of a theorem from the book (or the class notebook) concluding that g has at least one fixed point in the interval [1,2]. Give the theorem number, theorem part (Part (i), or (ii), or...) and book's edition (or show how to find it in the notebook) or simply state the part that matters to this question. And, clearly indicate how the hypotheses of that theorem are met. (Hint: g is continuous everywhere, except at x = -1.) (c) Find the best possible value of k, such that g'(x) ≤k, for all x € [1,2]. Show your reasoning clearly. (d) Find a theorem, or part of a theorem from the book (or the class notebook) concluding that 9 has exactly one fixed point in the interval [1, 2]. Give the theorem number, theorem part (Part (i), or (ii), or ...) and book's edition (or show how to find it in the notebook) or simply state the part that matters to this question. And, clearly indicate how the hypotheses of that theorem are met. (Hint: Does g' exist on the interval? Is k<1?) (e) Find the fixed point of g in the interval [1,2]. Show your work. (f) Find a theorem from the book (or the class notebook) concluding that the fixed-point iteration Xn+1 = g(xn) converges to the unique fixed point (found above) in the interval [1, 2]. Give the theorem name, or number and book's edition (or show how to find it in the notebook) or simply state the theorem. And, clearly indicate how the hypotheses of that theorem are met. g(x) = 1 + 1 1+x (a) Show that g maps the interval [1, 2] into the same interval [1, 2], which means to show that 1 ≤ g(x) ≤ 2, for all x € [1,2]. (b) Find a theorem, or part of a theorem from the book (or the class notebook) concluding that g has at least one fixed point in the interval [1,2]. Give the theorem number, theorem part (Part (i), or (ii), or...) and book's edition (or show how to find it in the notebook) or simply state the part that matters to this question. And, clearly indicate how the hypotheses of that theorem are met. (Hint: g is continuous everywhere, except at x = -1.) (c) Find the best possible value of k, such that g'(x) ≤k, for all x € [1,2]. Show your reasoning clearly. (d) Find a theorem, or part of a theorem from the book (or the class notebook) concluding that 9 has exactly one fixed point in the interval [1, 2]. Give the theorem number, theorem part (Part (i), or (ii), or ...) and book's edition (or show how to find it in the notebook) or simply state the part that matters to this question. And, clearly indicate how the hypotheses of that theorem are met. (Hint: Does g' exist on the interval? Is k<1?) (e) Find the fixed point of g in the interval [1,2]. Show your work. (f) Find a theorem from the book (or the class notebook) concluding that the fixed-point iteration Xn+1 = g(xn) converges to the unique fixed point (found above) in the interval [1, 2]. Give the theorem name, or number and book's edition (or show how to find it in the notebook) or simply state the theorem. And, clearly indicate how the hypotheses of that theorem are met. g(x) = 1 + 1 1+x

Answer rating: 100% (QA)

a To show that g maps the interval 1 2 into the same interval 1 2 we need to prove that 1 gx 2 for all x 1 2 Lets consider the function gx x 1x We want to show that for any x in the interval 1 2 gx is ... View the full answer