n be a positive integer. (a) Prove that if 0 < a < b, then a" < b". HINT: Use mathematical induction. (b) Prove that every nonnegative real number x has a unique nonnegative nth root x'/". HINT: The existence of x/" can be seen by applying the intermediate-value theorem to the function f(t) = t" for O The uniqueness follows from part (a)

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
icon
Related questions
Question
### Transcription for Educational Website

**Theorem on Positive Integers**

Let \( n \) be a positive integer.

#### (a) Inequality for Powers
Prove that if \( 0 \leq a < b \), then \( a^n < b^n \).

**Hint:** Use mathematical induction.

---

#### (b) Existence and Uniqueness of the nth Root
Prove that every nonnegative real number \( x \) has a unique nonnegative nth root \( x^{1/n} \).

**Hint:** The existence of \( x^{1/n} \) can be seen by applying the intermediate-value theorem to the function \( f(t) = t^n \) for \( t \geq 0 \). The uniqueness follows from part (a).
Transcribed Image Text:### Transcription for Educational Website **Theorem on Positive Integers** Let \( n \) be a positive integer. #### (a) Inequality for Powers Prove that if \( 0 \leq a < b \), then \( a^n < b^n \). **Hint:** Use mathematical induction. --- #### (b) Existence and Uniqueness of the nth Root Prove that every nonnegative real number \( x \) has a unique nonnegative nth root \( x^{1/n} \). **Hint:** The existence of \( x^{1/n} \) can be seen by applying the intermediate-value theorem to the function \( f(t) = t^n \) for \( t \geq 0 \). The uniqueness follows from part (a).
Expert Solution
steps

Step by step

Solved in 3 steps

Blurred answer
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning