Problem 69 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation \(x+e^{x}=0\) has (A) only one real root (B) only two real roots (C) no real root (D) None of these

Step-by-Step Solution

Answer

(A) only one real root

Step 1: Identify Function

Start by identifying the function that represents the equation. Here, the equation is given by \( f(x) = x + e^x \). Our task is to determine the real roots of this function, meaning the values of \( x \) for which \( f(x) = 0 \).

Step 2: Analyze the Function Behavior

We need to understand the behavior of \( f(x) = x + e^x \). The exponential function \( e^x \) is always positive for all real \( x \), and it increases as \( x \) increases. Adding \( x \) to \( e^x \) will generally make the function increase as \( x \) increases.

Step 3: Check Limits for Roots

Evaluate the function at certain points to establish limits. At \( x = 0 \), \( f(0) = 0 + e^0 = 1 \). Here, the function is greater than zero. Consider \( x \to - ext{{large}} \), as \( x \) becomes very negative, \( e^x \to 0 \) and \( x+0 \to x \), which approaches negative infinity. Thus, \( f(x) \) crosses the x-axis somewhere in the interval \( (- ext{{large}}, 0) \).

Step 4: Use Derivative for Monotonicity

Calculate the derivative \( f'(x) = 1 + e^x \). Since \( e^x \) is always positive, \( f'(x) = 1 + e^x \) is positive for all real \( x \). Therefore, \( f(x) \) is strictly increasing for all real \( x \).

Step 5: Conclusion on Real Roots

Given that the function \( f(x) \) is strictly increasing and crosses the x-axis at one point (as seen from the function behavior at infinite limits), \( f(x) = x + e^x \) can have only one real root.

Key Concepts

Exponential Function

The exponential function, specifically the form \( e^x \), plays a central role in mathematics. It is defined as a constant base \( e \) raised to the power of \( x \). Here, \( e \) is approximately 2.71828, a fundamental mathematical constant often known as Euler's number.
Exponential functions have some intriguing properties:

• They are always positive for all real values of \( x \); \( e^x > 0 \) for any real number \( x \).
• As \( x \) increases, \( e^x \) also increases. This is referred to as an increasing function.
• Exponential functions grow very quickly compared to polynomial functions as \( x \) becomes larger.

In the context of the equation \( x + e^x = 0 \), understanding \( e^x \) as always positive is key. It implies that even if \( x \) is negative, \( e^x \)'s contribution will be positive, affecting the overall behavior and solution of the equation.

Increasing Function

An increasing function is one that continuously rises as the input value (often written as \( x \)) increases. In mathematical terms, a function \( f(x) \) is increasing if for any two numbers \( a \) and \( b \), where \( a < b \), it holds that \( f(a) < f(b) \).
To determine if a function is increasing, you can:

• Consider the behavior of the function graphically; does the line move upwards as you move from left to right on a graph?
• Utilize calculus, specifically looking at the function's derivative.

In our equation \( f(x) = x + e^x \), the function is increasing because both components, \( x \) and \( e^x \), independently increase as \( x \) increases. As a result, their sum cannot decrease, making \( f(x) \) strictly increasing over all real numbers. This ensures that the function will only cross the x-axis once, giving it a single real root.

Derivative Analysis

Derivative analysis is a powerful technique in calculus used to determine the behavior and characteristics of functions. By taking the derivative of a function, you can find out where it is increasing or decreasing, the slope of the tangent, and even local maxima or minima.
To analyze the given function \( f(x) = x + e^x \), we must compute its derivative, denoted \( f'(x) \):

• Differentiate \( x \) to get 1, since the derivative of \( x \) with respect to itself is 1.
• Differentiate \( e^x \) to get \( e^x \), as the exponential function is unique with its derivative being the same as itself.

Thus, \( f'(x) = 1 + e^x \), which is always positive because \( e^x > 0 \) for all real \( x \).
This consistent positive value of \( f'(x) \) implies that \( f(x) \) is a strictly increasing function across the entire real line. Consequently, this indicates the function can have at most one real root, which is supported by the observation of its behavior at extreme values.