Problem 153 - 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: For \(b>a>1, \frac{1}{b \ln b}<\frac{f(b)-f(a)}{b-a}<\) \(\frac{1}{a \ln a}\), where \(f(x)=\ln (\ln x), x>1\) Reason: \(\frac{1}{x \ln x}\) is strictly decreasing in \((a, b)\)

Step-by-Step Solution

Answer

Option (A) is correct: Assertion(A) is True; Reason(R) is True and a correct explanation.

Step 1: Rewrite the Assertion

The assertion gives an inequality for the function \[f(x) = \ln (\ln x)\] in the interval \((a, b)\):\[ \frac{1}{b \ln b} < \frac{ f(b) - f(a) }{ b - a } < \frac{1}{a \ln a} \] A key task is proving or disproving this inequality based on the function and assumptions \(b > a > 1\). This will also involve using the Mean Value Theorem.

Step 2: Understand the Reason

According to the given Reason, the function \( \frac{1}{x \ln x} \) is strictly decreasing on the interval \((a, b)\). To verify this, we need to compute its derivative and check if it's negative within \((a, b)\).

Step 3: Calculate the Derivative for Reason

Consider \( h(x) = \frac{1}{x \ln x} \). The derivative will help establish whether it's decreasing:\[h'(x) = \frac{d}{dx} \left( \frac{1}{x \ln x} \right) = \frac{-(1 + \ln x)}{ (x \ln x)^2 }\] Since \(1 + \ln x > 0\) for \(x > 1\), \(h'(x) < 0\) implies \(h(x)\) is strictly decreasing.

Step 4: Apply Mean Value Theorem to Assertion

By Mean Value Theorem for \(f(x) = \ln(\ln x)\), there exists some \(c\) in \((a, b)\) such that:\[f'(c) = \frac{f(b) - f(a)}{b - a}\]Since \(f'(x) = \frac{1}{x \ln x}\), we have:\[\frac{1}{b \ln b} < f'(c) < \frac{1}{a \ln a}\] As \(f'(x) = \frac{1}{x \ln x}\) is decreasing, indeed, \(\frac{1}{b \ln b} < \frac{1}{c \ln c} < \frac{1}{a \ln a}\).

Step 5: Evaluate Conclusion for Assertion and Reason

Both the Assertion and Reason are true. Since \(\frac{1}{x \ln x}\) is decreasing, the function value between \(x = b\) and \(x = a\) satisfies the inequality, reinforcing that the Reason is a correct explanation for the Assertion.

Key Concepts

Calculus

Calculus is a branch of mathematics that focuses on change and motion. It allows us to study and understand quantities that vary over time. Calculus is divided into two main parts: differential and integral calculus. Both are essential for understanding concepts like rates of change and accumulation.

• Differential Calculus: Concerns the rate at which quantities change. It employs derivatives to determine slopes of functions which help in identifying behaviors of graphs.
• Integral Calculus: Focuses on accumulation of quantities, such as areas under curves and the summation of values over an interval. It uses integrals to accomplish this task.

In the context of the Mean Value Theorem, differential calculus plays a key role. It helps us understand how the assertion’s inequality can be explained by the behavior of derivatives.

Derivative

The derivative is a fundamental concept in calculus that represents the rate of change of a function with respect to a variable. It provides a linear approximation of the function at a given point and is crucial for analyzing functions.
To find the derivative of a function, we apply the principles of differential calculus.
In the provided exercise, the derivative of the function \( h(x) = \frac{1}{x \ln x} \) was computed to determine whether it is decreasing. The calculation was:- The derivative is given by \[h'(x) = \frac{d}{dx} \left( \frac{1}{x \ln x} \right) = \frac{-(1 + \ln x)}{ (x \ln x)^2 }\].- Because \(1 + \ln x > 0\) when \(x > 1\), the derivative \(h'(x)\) is negative, indicating \(h(x)\) is decreasing. This shows how derivatives help in studying the rate and direction of change of functions, which is necessary in applying the Mean Value Theorem in the assertion.

Inequality

An inequality is a mathematical statement that describes the relative size or order of two values. Inequalities often arise in dealing with functions, both in identifying their behaviors and proving various properties.
In the exercise, the assertion involves verifying an inequality for \(f(x) = \ln(\ln x)\):- \(\frac{1}{b \ln b} < \frac{f(b) - f(a)}{b - a} < \frac{1}{a \ln a}\) where \(b > a > 1\).Inequalities are useful because they set boundaries for the values that certain expressions can take. In calculus, inequalities are often tied to the behavior of derivatives and other function properties. The inequality in this context is confirmed by the derivative \(f'(x)\) being decreasing, and thus ensuring the mean value over the interval \((a, b)\) falls within the brackets given in the assertion.

Decreasing Function

A decreasing function is one where, as the input increases, the output either stays the same or decreases. In mathematical terms, a function \(f(x)\) is decreasing on an interval if for any two numbers \(x_1\) and \(x_2\) where \(x_1 < x_2\), we have \(f(x_1) \geq f(x_2)\).
This is key in the exercise because we showed that \(h(x) = \frac{1}{x \ln x}\) is a strictly decreasing function on the interval \((a, b)\).

• The derivative \(h'(x)\) is negative, indicating a decrease.
• This decreasing nature means that as \(x\) increases from \(a\) to \(b\), \(h(x)\) decreases, supporting the inequality given in the assertion.

Understanding decreasing functions is essential for interpreting the results of calculus operations, such as those dealing with the Mean Value Theorem and the specific behavior of functions in defined intervals.