QUESTION 1

Let 3', gr be continuous functions on [3, 4] and differentiable functions on [31 4]. We consider the function F given by F(:1:) = det(A) where ftm) vim) $2 + 4 A = f(3) 9(3) 13 f(4) 9(4) 20 {a} Your classmate Hesham asks you to find an s such that F[:1:) = [1. Enter your answer in the box below. One such value of :1: is :1: = @ fff; . {b} Your tutor Fenglin asks you to prove that there exists c E (3, 4) such that F' (c) = D. Provide your proof to your tutor Fenglin in the following essay box. Be sure to explain which theorems or results you use and how you use them. yntax advice: Use lvlaple's syntax for writing matrices. For example if your matrix 1 2 6 3 6 12 5 8 1 enter Equation A - A- Ix BIUSX X Styles Font . .. Words: 0 (b) What should be the third equation Saneer wrote down? Your answer should be in the Maple vector form such that ax + by + cz = d is the form of the third equation. Syntax advice: Note DO NOT include a comma for thousands of dollars in your answer. For the plane x - 2y + 3z = 4 you would type . Please use the "Preview" button to check your syntax. = (c) Determine the amount of each loan. Syntax advice: Write your answer in Maple vector form. Note DO NOT include a comma for thousands of dollars in your answer. Hint: Check you have the correct number of zeros for each loan amount in your entered solution and your solution satisfies at least the third equation Saneer wrote down from part (b). Please use the "Preview" button to check your syntax. Let A be a 5 x 5 matrix and suppose matrix A can be written as A = BC where B is a 5 x 4 matrix and C is a 4 x 5 matrix. (a) Does By = b, for b E Ro, have a unique, none and/or infinitely many solutions for y ? Select ALL options that apply. Unique None Infinitely many (b) Does Cx = 0, for 0 E R4 , have a unique, none and/or infinitely many solutions for x ? Select ALL options that apply. None Unique Infinitely many (c) Hence, or otherwise, explain in the essay box below, whether A is invertible or not. In your explanation you can use vec(x) and vec(0) to represent x and 0 respectively. UOneBEKO A - A- Ix BIUS X X Styles Font . .. Words: 0