Problem 155 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \((a+b)^{1 / n} \leq a^{1 / n}+b^{1 / n}\), where \(a, b \geq 0\) and \(n \geq 1\) Reason: The function \(f(x)=(1+x)^{p}-x^{p}-1, x \geq 0\) and \(0

Step-by-Step Solution

Answer

Option B: Assertion is True, Reason is True; Reason is not a correct explanation.

Step 1: Understand the Problem

We have an assertion and a reason, and we need to evaluate both to determine their truthfulness and relationship. **Assertion (A)**: \((a+b)^{1/n} \leq a^{1/n} + b^{1/n}\), where \(a, b \geq 0\) and \(n \geq 1\).**Reason (R)**: The function \(f(x)=(1+x)^{p}-x^{p}-1\), \(x \geq 0\) and \(0

Step 2: Analyze the Assertion

The assertion states that \((a+b)^{1/n} \leq a^{1/n} + b^{1/n}\), which is a version of the subadditivity property of power means when \(n \geq 1\). In the context where \(a,b \geq 0\), this inequality holds due to the concavity of the function \(x^{1/n}\). Therefore, the assertion is true.

Step 3: Analyze the Reason

The reason involves the function \(f(x) = (1+x)^p - x^p - 1\), with constraints \(0 < p \leq 1\) and \(x \geq 0\). To determine if \(f(x)\) is decreasing, we find its derivative, \(f'(x) = p((1+x)^{p-1} - x^{p-1})\). For \(0 < p \leq 1\), this derivative is non-positive, confirming that \(f(x)\) is indeed a decreasing function in \((0, \infty)\). Thus, the reason is true.

Step 4: Determine the Relation Between Assertion and Reason

Although both the assertion and the reason are individually correct, the reason does not actually explain the assertion. The reason discusses the decreasing nature of a particular function, but it is not linked to proving or explaining the subadditivity property described in the assertion. Thus, the reason is not a correct explanation for the assertion.

Key Concepts

Inequality

Inequalities are vital tools in mathematics, helping us understand relationships between different quantities. The assertion we deal with here, \((a+b)^{1/n} \leq a^{1/n} + b^{1/n}\), for \(a, b \geq 0\) and \(n \geq 1\), represents an inequality stemming from the concept of power means. In simple terms, this inequality expresses that the power mean of two numbers is less than or equal to the arithmetic mean of their respective roots.

The reason this inequality holds lies in the concavity of the function \(x^{1/n}\). A function is concave if, when you draw a line between any two points on its graph, the line does not pass above the graph. Concave functions result in subadditivity — a concept where combining things results in less than or equal to the sum of their parts. This is why the inequality is satisfied for non-negative numbers \(a\) and \(b\).

Understanding concavity and subadditivity is crucial in mastering inequalities. They are core concepts that help to determine how functions behave over particular ranges.

Power Mean

The power mean, often referred to as the generalized mean, is a versatile concept in statistics and mathematics. It extends the idea of the standard arithmetic mean to incorporate different powers, denoted by \(n\). The power mean of two numbers \(a\) and \(b\), with exponent \(n\), is given by \(((a^n + b^n)/2)^{1/n}\).

When considering different values of \(n\):

• For \(n=1\), it simplifies to the arithmetic mean, \((a+b)/2\).
• For \(n=0\), it corresponds to the geometric mean, \(\sqrt{ab}\).
• When \(n < 0\), larger numbers have less influence.

The inequality \((a+b)^{1/n} \leq a^{1/n} + b^{1/n}\) serves as a specific instance of the power mean inequality, where the function's concavity ensures that the mean of the roots is greater than or equal to the root of the mean. This demonstrates the ability of the power mean to capture diverse averaging methods based on the value of \(n\). It highlights how changing \(n\) influences the weight each number has in the mean.

Function Analysis

Function analysis is a branch of calculus that helps us understand the behavior and characteristics of functions, like the one presented in the reason: \(f(x)=(1+x)^p - x^p - 1\), where \(x \geq 0\) and \(0

For this function:

• The derivative \(f'(x) = p((1+x)^{p-1} - x^{p-1})\) shows us how \(f(x)\) changes as \(x\) changes.
• Since \(p((1+x)^{p-1} - x^{p-1}) \leq 0\) for \(0 < p \leq 1\), the function is decreasing in \((0, \infty)\).

This decrease indicates functions lose value, meaning they provide smaller outputs as inputs grow. Although the reason correctly depicts \(f(x)\) as decreasing, it doesn't explain the initial assertion, highlighting that each inequality and assertion builds upon different facets of mathematical reasoning.

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Problem 154

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Problem 156

Other exercises in this chapter

Problem 153

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Rea

View solution

Problem 154

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Rea

View solution

Problem 156

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Rea

View solution

Problem 157

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Rea

View solution

Key Concepts

Inequality

Inequalities are vital tools in mathematics, helping us understand relationships between different quantities. The assertion we deal with here, \((a+b)^{1/n} \leq a^{1/n} + b^{1/n}\), for \(a, b \geq 0\) and \(n \geq 1\), represents an inequality stemming from the concept of power means. In simple terms, this inequality expresses that the power mean of two numbers is less than or equal to the arithmetic mean of their respective roots.

The reason this inequality holds lies in the concavity of the function \(x^{1/n}\). A function is concave if, when you draw a line between any two points on its graph, the line does not pass above the graph. Concave functions result in subadditivity — a concept where combining things results in less than or equal to the sum of their parts. This is why the inequality is satisfied for non-negative numbers \(a\) and \(b\).

Understanding concavity and subadditivity is crucial in mastering inequalities. They are core concepts that help to determine how functions behave over particular ranges.

Power Mean

The power mean, often referred to as the generalized mean, is a versatile concept in statistics and mathematics. It extends the idea of the standard arithmetic mean to incorporate different powers, denoted by \(n\). The power mean of two numbers \(a\) and \(b\), with exponent \(n\), is given by \(((a^n + b^n)/2)^{1/n}\).

When considering different values of \(n\):

• For \(n=1\), it simplifies to the arithmetic mean, \((a+b)/2\).
• For \(n=0\), it corresponds to the geometric mean, \(\sqrt{ab}\).
• When \(n < 0\), larger numbers have less influence.

The inequality \((a+b)^{1/n} \leq a^{1/n} + b^{1/n}\) serves as a specific instance of the power mean inequality, where the function's concavity ensures that the mean of the roots is greater than or equal to the root of the mean. This demonstrates the ability of the power mean to capture diverse averaging methods based on the value of \(n\). It highlights how changing \(n\) influences the weight each number has in the mean.

Function Analysis

Function analysis is a branch of calculus that helps us understand the behavior and characteristics of functions, like the one presented in the reason: \(f(x)=(1+x)^p - x^p - 1\), where \(x \geq 0\) and \(0

For this function:

• The derivative \(f'(x) = p((1+x)^{p-1} - x^{p-1})\) shows us how \(f(x)\) changes as \(x\) changes.
• Since \(p((1+x)^{p-1} - x^{p-1}) \leq 0\) for \(0 < p \leq 1\), the function is decreasing in \((0, \infty)\).

This decrease indicates functions lose value, meaning they provide smaller outputs as inputs grow. Although the reason correctly depicts \(f(x)\) as decreasing, it doesn't explain the initial assertion, highlighting that each inequality and assertion builds upon different facets of mathematical reasoning.