Find the interval in which f(x)=cot^-1(log4 x) is strictly decreasing.

Name: Problem 28, Chapter 7: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-02-27T09:48:15-08:00
Duration: 1 min 14 s
Channel: Varsha Aggarwal
Description: Find the interval in which f(x)=cot^-1(log4 x) is strictly decreasing.

step 1 As we know that, as we know that, that core inverse x is strictly decreasing for all x in R, and

Find the interval in which f(x)=cot^-1(log4 x) is strictly...

Key Concepts

Domain and Range Considerations

When dealing with composite functions, it is vital to consider the domains and ranges of the individual functions involved. This ensures that the operations are valid and helps to identify the correct interval over which the function is defined and behaves in a specific manner, such as being strictly decreasing.

Logarithmic Functions

Logarithmic functions are introduced to transform multiplicative relationships into additive ones, and they have well-defined properties including specific bases and domains. In the context of composite functions, the characteristics of logarithmic functions, such as their increasing or decreasing behavior depending on the base, play a crucial role in determining the monotonicity of the overall function.

Function Composition and the Chain Rule

Many functions are defined as compositions of two or more functions. To analyze the behavior of such functions, one often uses the chain rule to differentiate them. Understanding how the inner function (in this case, a logarithmic function) and the outer function (an inverse trigonometric function) interact is essential for determining the intervals where the composite function is strictly decreasing.

Monotonicity

Monotonicity refers to the property of a function where it is consistently increasing or decreasing over an interval. Determining whether a function is strictly increasing or strictly decreasing typically involves analyzing the sign of its derivative. This concept is fundamental in calculus and is used to understand and predict the behavior of functions.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccot (cot?¹), are the reverse processes of trigonometric functions. They have specific domains and ranges, and their monotonicity properties are important when they appear in composite functions. Recognizing how these functions behave helps in understanding the overall function behavior when used as an outer function in compositions.

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