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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.

Calculus: Early Transcendentals
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...
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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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