Problem 82 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ y=\log _{3}\left(x^{2}-\sin x\right) $$

Step-by-Step Solution

Answer

There is no exact simple solution for this equation. The solution depends on the task we were to accomplish with this exercise. Could be to find the domain, or to find critical points, etc. The manipulation of this function requires understanding of both logarithmic properties and trigonometry.

Step 1: Understanding the Function

The function given is \( y=\log _{3}\left(x^{2}-\sin x\right) \). This equation is a logarithmic function based off base 3. The argument of the logarithmic function is the value \( x^{2}-\sin x \). This is a combination of a quadratic function and a sine function.

Step 2: Analyzing the Input of the Logarithm

The input of the logarithm (the value within the parentheses) must be greater than 0, because the logarithm of a negative value or zero is undefined. This means that \( x^{2}-\sin x \) must be greater than 0, i.e., \( x^{2}-\sin x > 0 \). This inequality restricts the values of x that can be used in the function. Depending on the exact task, this could be the domain of the function or could be used to find the x-intercept, among other things.

Step 3: Manipulating the Function and Finding the Derivative

We could also take derivative of the function as it could be useful for analyzing the function in more detail (for example, finding critical points, inflection points, intervals of increase/decrease, etc.). The derivative of the function is \( \frac{d}{dx} \log _{3}\left(x^{2}-\sin x\right) = \frac{ 2x - \cos x}{(\ln 3)(x^{2}-\sin x)} \)

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental concept in algebra and calculus, represented by the general form \( ax^2 + bx + c \). In our problem, the quadratic component is \( x^2 \), as part of the expression \( x^2 - \sin x \). Quadratic functions form a parabola when graphed. Understanding their properties helps in solving inequalities like \( x^2 - \sin x > 0 \). Let's delve into a few basic features:

• **Vertex**: The turning point of a quadratic graph. In its simplest form \( x^2 \), the vertex is at \( (0, 0) \).
• **Axis of Symmetry**: A vertical line passing through the vertex, typically \( x = -\frac{b}{2a} \) for general forms.
• **Direction of Opening**: The parabola opens upwards if \( a > 0 \), like in \( x^2 \), and downwards if \( a < 0 \).

Since our function includes \( x^2 \), it ensures that, overall, the expression tends to yield positive results for large values of \( x \). This interaction with \( \sin x \) is crucial when thinking about the intervals where \( x^2 - \sin x \) is greater than zero.

Sine Function

The sine function \( \sin x \) is one of the basic trigonometric functions, characterized by its wave-like oscillating behavior which ranges between -1 and 1. It introduces periodic and fluctuating elements to our mathematical expressions. In the expression \( x^2 - \sin x \), the \( \sin x \) part modulates the otherwise steadily increasing curve of \( x^2 \). Here's what you need to understand about \( \sin x \):

• **Periodicity**: It completes one full cycle over an interval of \( 2\pi \).
• **Amplitude**: The maximum deviation from the midline (0), with a range from -1 to 1.
• **Frequent Use**: Often used in modelling waves and oscillating systems due to its repetitive cycle.

In our function \( \sin x \) influences the lower edge of \( x^2 \) ensuring at points that \( x^2 \) may dip below threshold resulting in defining critical spots where the compound inequality \( x^2 - \sin x > 0 \) is valid.

Derivative of Functions

Derivatives are a cornerstone of calculus, measuring how a function changes as its input changes. They provide insights into the function's behavior, allowing us to explore its critical points and concavity. When we derive the logarithmic function \( y=\log_{3}(x^{2}-\sin x) \), we focus on how it changes with respect to \( x \). Let's break down the derivative:

• **Chain Rule**: Used when differentiating composite functions. First, find the derivative of the outer function (logarithm) applied to the inner function (\( x^2-\sin x \)).
• **Logarithm Rule**: When differentiating \( \log_{b}u \), the rule is \( \frac{1}{u \ln b} \cdot \frac{du}{dx} \).
• In our case: The derivative is \( \frac{ 2x - \cos x}{(\ln 3)(x^{2}-\sin x)} \).

This derivative helps us identify critical points by setting it equal to zero, intervals of increase or decrease by its sign, and any inflection by analyzing its behavior. Understanding these concepts is crucial for effectively interpreting and utilizing logarithmic functions in calculus.