#3. Thanks. 3. Use the MVT to show that the function f(x) = 1/x³ is uniformly continuous on the interval3,00). Hint: Mimic the proof of the uniform continuity of f(x) = T on the interval (0, 0)from the notes on Rolle and MVT theorems (Problem 12)%3D

Answered: Image /qna-images/answer/b6825bc6-ec5d-49af-8dbb-6ca942c700dc.jpg

Answered: 3. Use the MVT to show that the…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

See similar textbooks

Similar questions

What is the purpose of the Intermediate Value Theorem?
What does it mean to say that f(4) is a local maximum valueof f ?
Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.
7. If f(x) = (1-1²)et² dt, on which interval is f decreasing?
8. Show that the function f(2)= 2 is continuous everywhere but not differentiable anywhere.
3. Suppose f(x) = x ^ 4 - 4x ^ 2 + 6, and g(x) = 3x ^ 3 - 8x. Prove, via the Intermediate Value Theorem, that the functions intersect at least twice between x -2 and x = %3D 4. Suppose f(x) is continuous on the interval [0,1] with 0< = f(x)< = 1 for all x. Prove that f has a fixed point; that is, there exists a c in the interval [0,1] such that f(c) = c. (Hint: You need to use the Intermediate Value Theorem on the function h(x) = f(x) – x.)
2. a. f(x) = (x+2)/(x +4). Use the Mean Value Theorem on the interval [0, 4] to find all values of x in the interval (0, 4) such that the value of the first derivative of f at those points is (f(b) – f(a))/(b – a). b. For the interval [-5, 0], fb) - fa))(b — а)-GG0) — f-5)(0 — (-5)) = (½ – 3)/(0 + 5) = -½ but, since the first derivative of f(x) is always positive, nowhere on that interval does the first derivative of f(x) = -½. Thus the Mean Value Theorem does not hold for f(x) on this interval. Why?
6. Use the Mean Value Theorem to find all values of c in (6, 14) such that ƒ'(c) = f(b)-f(a) b-a for f(x)=√√√2x - 3 on the interval [6, 14]
2. A continuous function f is defined on the interval [-2, 2]. The values of f at some of the points of the interval are given by the following table: -2 -1 1 2 |f(x) 2 -1 2 -1 2 (a) Using only this information, what can be concluded about the roots of f, that is, the solutions of f(x) = 0, in the interval [-2, 2]? The answer should be something like: f has at least 8 roots in [-2,2], or ƒ has at most 6 roots in [-2, 2]. Hint: Use the Intermediate Value Theorem on each of the intervals [-2, –1], [-1,0], [0,1], and [1, 2]. (b) If f(x) = xª – 4.x² + 2, verify that the relevant values of f are given by the table above. (a) Sketch the graph of y = f(x) in the viewing window [-2.5, 2.5]×[-3,3]. (b) How many roots does f have in the interval [-2,2]? Find the roots algebraically. Suggestion: Let t = x? and solve with the quadratic formula. Then find x. (c) If f(x) = x+ – 4x? + 2+ 5(2x – 1)x(x² – 1)(x² – 4). Verify that the relevant values of f are given by the table above. (a) Sketch the…

Recommended textbooks for you

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

College Algebra (MindTap Course List)

Algebra

ISBN:

9781305652231

Author:

R. David Gustafson, Jeff Hughes

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

College Algebra (MindTap Course List)

Algebra

ISBN:

9781305652231

Author:

R. David Gustafson, Jeff Hughes

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

College Algebra

Algebra

ISBN:

9781305115545

Author:

James Stewart, Lothar Redlin, Saleem Watson

Publisher:

Cengage Learning

Algebra: Structure And Method, Book 1

Algebra

ISBN:

9780395977224

Author:

Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:

McDougal Littell

Trigonometry (MindTap Course List)

Trigonometry

ISBN:

9781337278461

Author:

Ron Larson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

3. Use the MVT to show that the function f(x) = 1/x³ is uniformly continuous on the interval 3,00). Hint: Mimic the proof of the uniform continuity of f(x) = T on the interval (0, 0) from the notes on Rolle and MVT theorems (Problem 12) %3D