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60. The geometric, logarithmic, and arithmetic mean inequality a. Show that the graph of e* is concave up over every interval of x-values. b. Show, by reference to the accompanying figure, that if 0 < a < b then Inb elna + elnb (Ina+Inb)/2. (Inb – Ina) < • (In b – In a). e* dx < 2 Ina y = e* B х In a + In b In b In a 2 NOT TO SCALE c. Use the inequality in part (b) to conclude that b – a a + b Vab Inb – Ina 2 This inequality says that the geometric mean of two positive numbers is less than their logarithmic mean, which in turn is less than their arithmetic mean.

60. The geometric, logarithmic, and arithmetic mean inequality a. Show that the graph of e* is concave up over every interval of x-values. b. Show, by reference to the accompanying figure, that if 0 < a < b then Inb elna + elnb (Ina+Inb)/2. (Inb – Ina) < • (In b – In a). e* dx < 2 Ina y = e* B х In a + In b In b In a 2 NOT TO SCALE c. Use the inequality in part (b) to conclude that b – a a + b Vab Inb – Ina 2 This inequality says that the geometric mean of two positive numbers is less than their logarithmic mean, which in turn is less than their arithmetic mean.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 23E
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60. The geometric, logarithmic, and arithmetic mean inequality a. Show that the graph of e* is concave up over every interval of x-values. b. Show, by reference to the accompanying figure, that if 0 < a < b then Inb elna + elnb (Ina+Inb)/2. (Inb – Ina) < • (In b – In a). e* dx < 2 Ina y = e* B х In a + In b In b In a 2 NOT TO SCALE c. Use the inequality in part (b) to conclude that b – a a + b Vab Inb – Ina 2 This inequality says that the geometric mean of two positive numbers is less than their logarithmic mean, which in turn is less than their arithmetic mean.
Transcribed Image Text:60. The geometric, logarithmic, and arithmetic mean inequality a. Show that the graph of e* is concave up over every interval of x-values. b. Show, by reference to the accompanying figure, that if 0 < a < b then Inb elna + elnb (Ina+Inb)/2. (Inb – Ina) < • (In b – In a). e* dx < 2 Ina y = e* B х In a + In b In b In a 2 NOT TO SCALE c. Use the inequality in part (b) to conclude that b – a a + b Vab Inb – Ina 2 This inequality says that the geometric mean of two positive numbers is less than their logarithmic mean, which in turn is less than their arithmetic mean.
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