Answer to Question #153784 in Calculus for rohan

Question #153784

using suitable function f(x), obtain the following inequality for a < b < 1 using mean value theorem


1
Expert's answer
2021-01-05T16:40:56-0500

Consider f(x)= In x, so we apply the Lagrange Formula to it on the interval [a,b]

"\\frac{f(b)-f(a)}{b-a}=f'(c)" where c "\\in(a, b)\\\\"

Hence

"\\frac{Inb-Ina}{b-a}=\\frac{1}{c}\\\\\nIn\\frac{b}{a}=(b-a)\\frac{1}{c}\\\\"

The right side of the equation is minimal when c = b and respectively takes a maximum at c = a, as a result, we get the inequality.

"\\frac{b-a}{b}\\le In\\frac{b}{a}\\le\\frac{b-a}{a}"


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