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

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

The equation has only one real root (option A).

Step 1: Analyzing Function Behavior

Consider the function \( f(x) = x + e^x \). We need to analyze its behavior to see where it might intersect with the x-axis (i.e., where \( f(x) = 0 \)).

Step 2: Evaluate Function at Critical Points

Let's evaluate the function at some critical points. For \( x = 0 \), \( f(0) = 0 + e^0 = 1 \), which is positive. As \( x \to -\infty \), \( e^x \to 0 \) and \( x \to -\infty \), so \( f(x) \to -\infty \). This suggests the function crosses the x-axis somewhere in \( x < 0 \).

Step 3: Apply Intermediate Value Theorem

Since \( f(x) \) is continuous and \( f(x) \to -\infty \) as \( x \to -\infty \) while \( f(0) = 1 \), the Intermediate Value Theorem indicates there must be at least one root for some \( x < 0 \).

Step 4: Check for Multiple Roots

To determine if there are multiple roots, consider the derivative \( f'(x) = 1 + e^x \). Since \( e^x > 0 \) for all real \( x \), \( f'(x) > 0 \). This means \( f(x) \) is strictly increasing, hence it cannot have more than one root because a strictly increasing function crosses the x-axis at most once.

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that helps us establish the existence of roots within a certain interval. This theorem is applicable only to continuous functions, making it a perfect fit for our function, \( f(x) = x + e^x \). Here's why it's important:

• Continuity necessity: The function \( f(x) = x + e^x \) is continuous because both \( x \) and \( e^x \) are continuous functions. Therefore, the sum is also continuous.
• The existence of a root: According to the IVT, if \( f(x) \) takes on opposite signs at two points \( a \) and \( b \) (i.e., \( f(a) < 0 \) and \( f(b) > 0 \) or vice versa), there exists at least one real root \( c \) in the interval \( (a, b) \) where \( f(c) = 0 \).
• Application in our problem: Evaluating \( f(x) \) at critical points shows that \( f(x) \to -\infty \) as \( x \to -\infty \), and \( f(0) = 1 \), indicating a sign change and the presence of at least one root in \( x < 0 \).

Derivative Analysis

Derivative analysis helps us comprehend the behavior of a function, such as identifying whether it's increasing or decreasing. In this case, we examine the derivative of \( f(x) = x + e^x \), which is \( f'(x) = 1 + e^x \).

• Derivative calculation: The derivative, \( f'(x) = 1 + e^x \), is always greater than zero because \( e^x \) is always positive for any real value of \( x \). This means \( 1 + e^x \) is positive as well.
• Strictly increasing behavior: Since \( f'(x) > 0 \) for all \( x \), the function \( f(x) \) is strictly increasing across its entire domain. This is crucial because it indicates the function never decreases or flattens out, thus ensuring that \( f(x) \) can cross the x-axis only once.
• Implication for root: The fact that the function is strictly increasing implies that if there is a root, there cannot be another one. Once \( f(x) \) crosses the x-axis, it will keep increasing away from it.

Exponential Functions

Exponential functions are a key element in understanding the behavior of the provided function \( f(x) = x + e^x \). They have unique characteristics that make them both interesting and challenging.

• Rapid growth: The key feature of exponential functions like \( e^x \) is their rapid rate of increase. As \( x \) becomes more positive, \( e^x \) increases exponentially, which means it grows much faster than polynomial functions.
• Always positive: For all real numbers \( x \), \( e^x \) is always positive. This is essential when analyzing derivatives and function behavior, as seen with \( f'(x) = 1 + e^x \).
• Influence on \( f(x) \): In \( f(x) = x + e^x \), the exponential component, \( e^x \), dominates for large positive \( x \), contributing to a rapid increase in function value. Conversely, for large negative \( x \), \( e^x \) approaches zero, allowing the linear term \( x \) to dominate.