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(i) Using the facts that √ and 1/x are, respectively, increasing and decreasing functions of a real variable x > 0, show that f(x) = √√x(1 – 1) is an increasing function of x > 0. Conclude that f(x) > 1 for all x ≥ 3.

(i) Using the facts that √ and 1/x are, respectively, increasing and decreasing functions of a real variable x > 0, show that f(x) = √√x(1 – 1) is an increasing function of x > 0. Conclude that f(x) > 1 for all x ≥ 3.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.6: Inequalities
Problem 79E
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(i) Using the facts that √x and 1/x are, respectively, increasing and decreasing functions of a real variable x > 0, show that f(x) = √(1-1) is an increasing function of x > 0. Conclude that f(x) > 1 for all x ≥ 3. (ii) Show that the inequality ö(p²) > p¹/² holds for every prime power of an odd prime p. Conclude that 4(m) ≥ √m, for every odd m. Problem Notes. One shows with very little additional effort that for even m the inequality 4(m) ≥ √/m/2 holds. These inequalities are far from the best that can be proved but they are sufficient to prove the following interesting result. Proposition. For each fixed integer n, the equation y(x) = n has only a finite number of solutions (possibly none). This isn't part of the test.
Transcribed Image Text:(i) Using the facts that √x and 1/x are, respectively, increasing and decreasing functions of a real variable x > 0, show that f(x) = √(1-1) is an increasing function of x > 0. Conclude that f(x) > 1 for all x ≥ 3. (ii) Show that the inequality ö(p²) > p¹/² holds for every prime power of an odd prime p. Conclude that 4(m) ≥ √m, for every odd m. Problem Notes. One shows with very little additional effort that for even m the inequality 4(m) ≥ √/m/2 holds. These inequalities are far from the best that can be proved but they are sufficient to prove the following interesting result. Proposition. For each fixed integer n, the equation y(x) = n has only a finite number of solutions (possibly none). This isn't part of the test.
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