Problem 127 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\((1+x)^{p} \leq 1+x^{p}\), where (A) \(p>1\) (B) \(0 \leq p \leq 1\) (C) \(x>0\) (D) \(x<0\)

Step-by-Step Solution

Answer

The inequality holds only when \( 0 \leq p \leq 1 \).

Step 1: Analyzing option (A) with p > 1

For the case when \( p > 1 \), consider expanding \( (1 + x)^p \) using the binomial theorem:\[ (1 + x)^p = 1 + px + \frac{p(p-1)}{2}x^2 + \ldots \] Since \( p > 1 \), all terms in this expansion are positive when \( x > 0 \). Consequently, \( (1 + x)^p > 1 + x^p \) because the additional terms in the expansion make it larger than \( 1 + x^p \). Therefore, the inequality \( (1+x)^{p} \leq 1+x^{p} \) does not hold for \( p > 1 \).

Step 2: Analyzing option (B) with 0 <= p <= 1

For \( 0 \leq p \leq 1 \), consider the function \( f(x) = (1 + x)^p - (1 + x^p) \) and its derivative:\[ f'(x) = p(1+x)^{p-1} - px^{p-1} \]For \( 0 \leq p \leq 1 \), the derivative is less than or equal to zero for any \( x \geq 0 \), indicating that \( f(x) \) is non-increasing for \( x \geq 0 \). Therefore, the inequality \( (1+x)^p \leq 1 + x^p \) holds when \( 0 \leq p \leq 1 \).

Step 3: Checking condition (C) for x > 0

For \( x > 0 \), consider the previous analysis. In step 1, we found that for \( p > 1 \), the inequality \( (1+x)^{p} > 1+x^{p} \) holds because additional terms in the expansion contribute to a value greater than \( 1 + x^p \). Thus, the inequality does not hold when \( x > 0 \) and \( p > 1 \), but it holds when \( 0 \leq p \leq 1 \).

Step 4: Checking condition (D) for x < 0

For \( x < 0 \), observe that \( x^p \) can be complex or undefined if \( p \) is not an integer, making direct comparison challenging. However, if \( x^p \) is real and defined for specific \( p \), the inequality’s nature could change. Generally, when \( x < 0 \) and \( p > 1 \), \( (1+x)^p \) contributes to positive components offset negatively, but when \( 0 \leq p \leq 1 \), the inequality holds because the function \( (1 + x) \), raised to a smaller power, permits \( (1 + x^p) \) to remain larger or equal.

Key Concepts

Understanding Binomial Theorem

The binomial theorem is a powerful tool used to expand expressions that are raised to a power. It states that for any positive integer \( n \), \\[ (1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots + x^n \] \This provides a way to open up an expression like \((1+x)^p\). In the context of the given inequality problem, using the binomial theorem helps identify additional positive terms when \( p > 1 \) which make the left-hand side greater than the right-hand side, i.e., \( 1 + x^p \). \

• Each term in the expansion represents a higher power of \(x\), with coefficients becoming progressively larger.
• The theorem is particularly useful in inequalities like our exercise.
• The terms added by the expansion are significant to determine if the inequality holds for given \( p \) values.

Exploring Derivative Analysis

Derivative analysis involves studying the behavior of a function by examining its derivative. Here, we have the function \( f(x) = (1 + x)^p - (1 + x^p) \) and its derivative\\[ f'(x) = p(1+x)^{p-1} - px^{p-1} \] \The derivative tells us how the function \( f(x) \) changes as \( x \) changes. If the derivative is non-positive, it means the function is non-increasing. \

• For \( 0 \leq p \leq 1 \), the derivative is less than or equal to zero, suggesting a non-increasing function over non-negative \( x \).
• This analysis helps us conclude that the inequality \( (1+x)^p \leq 1 + x^p \) holds when \( p \) is between 0 and 1 inclusive.

\Using derivatives is a strategic approach to efficiently determine the intervals where the inequality holds.

Functional Inequality Analysis

Functional inequality analysis involves checking how different parts of an inequality behave under given conditions. In our exercise, we're comparing \( (1+x)^p \) to \( 1+x^p \). By analyzing \( f(x) = (1 + x)^p - (1 + x^p) \), we can determine conditions under which the inequality holds. \

• For \( x > 0 \), if \( p > 1 \), the expanded expression using binomial theorem makes \( (1+x)^p \) larger than \( 1+x^p \).
• For \( x < 0 \), the situation changes as the nature of \( x^p \) can become complex, especially when \( p \) is not an integer.
• When \( 0 \leq p \leq 1 \), the function tends to keep \( (1+x)^p \leq 1 + x^p \), especially for \( x \geq 0 \).

\Functional inequality analysis is thus an essential method to decode the behavior of mathematical functions under specific conditions.

Understanding the Conditions for Inequalities

Conditions for inequalities are essential to solve problems like ours by understanding under what conditions the inequality holds true. In our exercise, this means recognizing the relationship between \( p \) and \( x \). \

• For \( p > 1 \), \( (1+x)^p \) generally exceeds \( 1+x^p \) when \( x > 0 \).
• For \( 0 \leq p \leq 1 \), the inequality holds more comfortably because of the nature of the power and expansion effects, especially when \( x \) is non-negative.
• If \( x < 0 \), different rules might apply because of potential complex numbers in \( x^p \).

\By understanding these conditions, one can determine when and why an inequality will hold or fail based on its components. This fundamental understanding helps in various branches of mathematics, particularly in solving equations under constraints.