Problem 139 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x)\) be a function of \(x\), where \(f(x)\) is continuous in the closed interval \([a, b]\) and differentiable in the open interval \((a, b)\). Also, \(f(a)=f(b)\), i.e., the values at the end points \(a\) and \(b\) are equal. Then, there exists at least one point \(c\) between \(a\) and \(\mathrm{b}\) (i.e., \(a

Step-by-Step Solution

Answer

The answer is (A) at least one root.

Step 1: Analyze the Given Information

The given problem involves two equations: \(e^{x} \sin x = 1\) and \(e^{x} \cos x = -1\). We are to determine how many roots of the second equation exist between any two real roots of the first equation. This is essentially asking how many times, between any two roots of the first equation, the derivative of a related function is zero.

Step 2: Apply Rolle's Theorem

Rolle's Theorem states that if a function \(f\) is continuous on a closed interval \([a, b]\) and differentiable on \((a, b)\), and \(f(a) = f(b)\), then there exists at least one \(c\) in \((a, b)\) such that \(f'(c) = 0\). We will use this to analyze the functions related to the given equations.

Step 3: Construct the Function for Rolle's Application

Consider the function \(f(x) = e^{x} \sin x\). We know that \(f(x_1) = e^{x_1} \sin x_1 = 1\) and \(f(x_2) = e^{x_2} \sin x_2 = 1\) for any two consecutive roots \(x_1\) and \(x_2\) of \(e^{x} \sin x = 1\). Thus, the conditions of Rolle's Theorem are satisfied on the interval \([x_1, x_2]\).

Step 4: Evaluate the Derivative

The derivative \(f'(x)\) involves differentiating \(e^{x} \sin x\), which gives \(f'(x) = e^{x} \cos x + e^{x} \sin x\). Setting \(f'(x) = 0\) gives \(e^{x} \cos x + e^{x} \sin x = 0\), which simplifies to \(e^{x} \cos x = -e^{x} \sin x\). Simplifying further, we have \(e^{x} \cos x = -1\). Thus, between any two roots of \(e^{x} \sin x = 1\), the equation \(e^{x} \cos x = -1\) must have at least one root.

Step 5: Conclusion

By Rolle's Theorem and the previous steps, there is guaranteed to be at least one root of \(e^{x} \cos x = -1\) between any two consecutive roots of \(e^{x} \sin x = 1\). Therefore, the correct option is (A) at least one root.

Key Concepts

Differential Calculus

Differential calculus is a fundamental branch of calculus that focuses on the rate at which quantities change. It is primarily concerned with the concept of the derivative, which represents the rate of change of a function concerning its variable. This means if you have a function describing how a quantity changes over time or space, the derivative describes how fast that change is happening at any particular point.

The main tool in differential calculus is the derivative, which provides information about the slope of the tangent line to the function at any point. This concept is crucial when analyzing any function to determine its behavior, such as finding maxima, minima, and points of inflection. In the exercise problem, understanding differential calculus is essential when using Rolle's Theorem to find when the derivative of a function equals zero, indicating a horizontal tangent to the curve.

Continuity

Continuity is a critical concept in calculus, specifically in discussions of differential calculus and the application of certain theorems like Rolle's Theorem. A function is continuous on an interval if there are no 'breaks', 'holes', or 'jumps' in its graph over that interval. Mathematically, a function \(f(x)\) is continuous at a point \(c\) if the limit of \(f(x)\) as \(x\) approaches \(c\) equals the function value \(f(c)\).

A continuous function over a closed interval \([a, b]\) means each point on the interval has a defined value, and there are no unpredicted jumps between those points. This foundation was crucial for applying Rolle's Theorem in the given exercise, as the theorem requires both endpoint continuity and differentiability in the open interval to ensure that a derivative equals zero somewhere within.

Derivatives

Derivatives are one of the foundational aspects of differential calculus. They represent the instantaneous rate of change of a function with respect to one of its variables. For a function \(f(x)\), the derivative is often represented as \(f'(x)\) or \(\frac{df}{dx}\).

In practical terms, a derivative at a specific point tells how "steep" the function is at that point. If \(f'(x) = 0\), it indicates a horizontal tangent line at that point, suggesting potential local maxima or minima. This property is leveraged in the exercise problem, where the zeroes of the derivative \(f'(x)\) indicate points between roots of a related function, solving the problem using Rolle's Theorem.

Roots of Equations

Roots of an equation refer to values of \(x\) that make the equation equal to zero. Finding these roots involves solving the equation and locating points where the function crosses the x-axis.

In the context of the exercise, the roots of the equation \(e^{x} \sin x = 1\) are important because they form the foundation for applying Rolle's Theorem. Between any two such roots, the distinct behavior of the function's derivative allows for the identification of at least one root of \(e^{x} \cos x = -1\), as confirmed through the theorem's application. The ability to find and understand these roots can lead to a deeper comprehension of the function's behavior and characteristics.