Problem 13 - 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: Analyze and Understand the Function

We need to analyze the function given by the equation \( f(x) = x + e^{x} \). Our goal is to determine where this function equals zero and whether it happens at real values of \( x \).

Step 2: Consider Behavior as \( x \to - finfty \)

As \( x \to -fty \), the term \( e^x \to 0^+ \) because \( e^x \) is an exponential function that approaches zero from the positive side. Thus, \( f(x) = x + e^x \approx x \) and approaches \( -fty \) as \( x \to -fty \).

Step 3: Consider Behavior as \( x \to + fty \)

As \( x \to +fty \), the term \( e^x \to +fty \). Therefore, \( f(x) = x + e^x \) approaches \( +fty \) because \( e^x \) dominates \( x \).

Step 4: Evaluate the Function at Specific Points

Calculating the value at \( x = 0 \), we have \( f(0) = 0 + e^0 = 1 \). And, let's test a negative value, say \( x = -1 \). We find \( f(-1) = -1 + e^{-1} \approx -1 + 0.3679 = -0.6321 \). These evaluations suggest that there is a crossing between negatives values and zero.

Step 5: Employ Intermediate Value Theorem

Since \( f(x) \) is continuous (as a sum of continuous functions) on \( \mathbb{R} \), and \( f(-1) < 0 \) while \( f(0) > 0 \), there exists at least one real root in the interval \((-1, 0)\).

Step 6: Analyze the Derivative for Uniqueness

Differentiating \( f(x) = x + e^x \) gives \( f'(x) = 1 + e^x \). Since \( e^x \) is always positive for real \( x \), \( f'(x) > 0 \) for all \( x \). This indicates that \( f(x) \) is strictly increasing, meaning it can cross the x-axis at most once.

Key Concepts

Continuity

The concept of continuity is crucial when analyzing functions and their behaviors. In simple terms, a function is continuous if you can draw its graph without lifting your pen from the paper. For the function given in our original exercise, which is defined as \( f(x) = x + e^x \), we note that it comprises the sum of two basic functions: a linear function \( x \) and an exponential function \( e^x \). Both of these functions are continuous over all real numbers.

Therefore, their sum \( f(x) \) is also continuous over all real numbers.

• Linear function \( x \): It's simply a straight line and is continuous everywhere.
• Exponential function \( e^x \): Smooth and continuous, defined for all \( x \).

Understanding this continuity allows us to infer that there are no interruptions or "gaps" in the graph of \( f(x) \).

This continuity is vital because it supports the application of other mathematical tools, such as the Intermediate Value Theorem, which depends on a function being continuous over a specific interval.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a central concept in calculus. It helps find roots of functions when we know two points on a continuous curve, one above the x-axis and one below. The theorem states that if a function is continuous on a closed interval \([a, b]\) and takes on different signs at the endpoints of the interval (i.e., \( f(a) \) and \( f(b) \) are of opposite signs), then there exists at least one \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \).

This concept is applied in our original exercise by evaluating \( f(-1) \) and \( f(0) \):

• \( f(-1) = -1 + e^{-1} \approx -0.6321 \), which is negative.
• \( f(0) = 0 + e^{0} = 1 \), which is positive.

Because \( f(-1) < 0 \) and \( f(0) > 0 \), and knowing \( f(x) \) is continuous (from our previous discussion), the IVT guarantees that there is at least one real root of the equation \( f(x) = 0 \) somewhere in the interval \((-1, 0)\).

The IVT is a powerful and practical tool for solving equations and understanding the behavior of continuous functions over real intervals.

Differentiation

Differentiation is a process in calculus used to find the rate at which a function changes. It reveals critical aspects of the function's behavior, like where it increases or decreases. The differentiation of our function \( f(x) = x + e^x \) results in the derivative \( f'(x) = 1 + e^x \).

By examining this derivative, we can determine:

• Positivity: Since \( e^x \) is always positive for real \( x \), the derivative \( f'(x) \) is greater than zero everywhere.
• Increasing Nature: \( f(x) \) is strictly increasing across its entire domain because the derivative does not switch signs.

For our specific exercise, the fact that \( f(x) \) is always increasing implies it will cross the x-axis at most once. If it starts increasing from a negative value and becomes positive, it means that \( f(x) \) can have only one real root.

Differentiation thus becomes a crucial part of verifying the uniqueness of a possible real root. It ensures there are no multiple crossings or turning points, which maintains that our function's real root solution to \( f(x) = 0 \) is unique.