(B) Let $f(x) = \frac{\log(\pi + x)}{\log(e + x)}$.Using the quotient rule,$f'(x) = \frac{\log(e + x) \cdot \frac{1}{\pi + x} - \log(\pi + x) \cdot \f

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(B) Let $f(x) = \frac{\log(\pi + x)}{\log(e + x)}$.Using the quotient rule,$f'(x) = \frac{\log(e + x) \cdot \frac{1}{\pi + x} - \log(\pi + x) \cdot \frac{1}{e + x}}{[\log(e + x)]^2}$.$f'(x) = \frac{(e + x)\log(e + x) - (\pi + x)\log(\pi + x)}{(\pi + x)(e + x)[\log(e + x)]^2}$.Since $e 0$,we have $e + x Thus,$(e + x)\log(e + x) Therefore,$f(x)$ is a decreasing function on $(0, \infty)$.Hence,option $B$ is the correct answer.

Class 12 Mathematics Applications of Derivatives Increasing and Decreasing function

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What kind of function is $f(x) = \frac{\log(\pi + x)}{\log(e + x)}$?

A
Increasing on $(0, \infty)$
B
Decreasing on $(0, \infty)$
C
Increasing on $(0, \frac{\pi}{e})$ and decreasing on $(\frac{\pi}{e}, \infty)$
D
Decreasing on $(0, \frac{\pi}{e})$ and increasing on $(\frac{\pi}{e}, \infty)$

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Applications of Derivatives

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Increasing and Decreasing function

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What kind of function is $f(x) = \frac{\log(\pi + x)}{\log(e + x)}$?

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