The funetion f(x)=(log(π+x))/(log(e+x)) is (a) increasing on [0, ∞) (b) decreasing on [0, ∞) (c)

step 1 in this problem to find whether a function is increasing or decreasing in an interval we'll find

The funetion f(x)=(log(π+x))/(log(e+x)) is (a) increasing on...

The funetion $f(x)=\frac{\log (\pi+x)}{\log (e+x)}$ is
(a) increasing on $[0, \infty)$
(b) decreasing on $[0, \infty)$
(c) increasing on $\left[0, \pi^{\prime} e\right)$ and decreasing on $[\pi e, \infty)$
(d) decreasing on $[0, \pi e)$ and increasing on $[\pi l e, \infty)$

Key Concepts

Chain Rule

The chain rule is used for differentiating composite functions by accounting for the derivative of the inner function. It is particularly important when dealing with functions that have shifted inputs or nested expressions in the argument of another function, ensuring correct application of the derivative.

Quotient Rule

The quotient rule is a technique in calculus used to differentiate functions expressed as the ratio of two differentiable functions. It is crucial for handling cases where both the numerator and the denominator contribute to the overall rate of change, such as in functions that involve ratios of logarithms.

Monotonicity of Functions

Monotonicity refers to whether a function is consistently increasing or decreasing over an interval. Determining monotonicity is essential for understanding the behavior of functions and involves analyzing the sign of the derivative across the domain.

Differentiation and the Derivative Test

Differentiation is a fundamental tool used to compute the rate of change of a function. The derivative test, which examines where the derivative is positive or negative, is pivotal in establishing whether a function is increasing or decreasing over a specified interval.

Logarithmic Functions

Logarithmic functions are a common type of function in mathematics, characterized by their logarithm operation. They have specific properties such as defined domains (typically positive real numbers) and derivatives that involve the reciprocal of their argument, making them an important component in various analytical problems.

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