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Differential Calculus> Image attached Show by LMVT and taking f(...

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Image attached Show by LMVT and taking f(x) = log x that ( question attached in image)

Drake , 10 Years ago

Grade 12

1 Answers

Jitender Singh

Last Activity: 10 Years ago

Ans:

$f(x) = log(x)$

Interval = [a, b]

Let ‘c’ be the point in the interval.

Then, by LMVT

$f'(c) = \frac{f(b)-f(a)}{b-a}$

$f'(c) = \frac{log(b)-log(a)}{b-a}$

$f'(x) = \frac{1}{x}$

$f''(x) = \frac{-1}{x^{2}}<0$

-> f’(x) is always decreasing. So, we have

$f'(b) < f'(c) < f'(a)$

$\frac{1}{b} < \frac{log(\frac{b}{a})}{b-a} < \frac{1}{a}$

$\frac{b-a}{b} < log(\frac{b}{a}) < \frac{b-a}{a}$

$1-\frac{a}{b} < log(\frac{b}{a}) < \frac{b}{a}-1$

Thanks & Regards

Jitender Singh

IIT Delhi

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