5. Prove that there is no value of k such that the equation æ³ – 3x + k = 0 has two distinct roots in [0, 1]. Hint: Let f(x) = x³ – 3+ k. Assume there are two distinct a, b e (0, 1] such that f(a) = f(b) = 0. Apply Rolle's theorem to f on the interval (a, b] to conclude that there exists a c such that f'(c) = 0. This should lead to a contradiction.

5. Prove that there is no value of k such that the equation æ³ – 3x + k = 0 has two distinct roots in [0, 1]. Hint: Let f(x) = x³ – 3+ k. Assume there are two distinct a, b e (0, 1] such that f(a) = f(b) = 0. Apply Rolle's theorem to f on the interval (a, b] to conclude that there exists a c such that f'(c) = 0. This should lead to a contradiction.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 30EQ

See similar textbooks

Similar questions

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xy
Let f(x),g(x),h(x)F[x] where f(x) and g(x) are relatively prime. If h(x)f(x), prove that h(x) and g(x) are relatively prime.
For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.
18. Let and be defined as follows. In each case, compute for arbitrary . a. b. c. d. e.
31. Prove statement of Theorem : for all integers and .