Problem 67 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(f\) be a differentiable function such that \(f^{\prime}(x)=7-\frac{3}{4} \frac{f(x)}{x}\), \((x>0)\) and \(f(1) \neq 4\). Then \(\lim _{x \rightarrow 0^{+}} x f\left(\frac{1}{x}\right)\) (a) exists and equals \(\frac{4}{7}\). (b) exists and equals 4 . (c) does not exist. (d) exists and equals 0 .

Step-by-Step Solution

Answer

Option (a) exists and equals \(\frac{4}{7}\).

Step 1: Identify the given function and conditions

We are given that the derivative of the function is \(f'(x) = 7 - \frac{3}{4} \frac{f(x)}{x}\) for \(x > 0\), and \( f(1) eq 4\). We need to find the behavior of \(x f\left(\frac{1}{x}\right)\) as \( x \to 0^+ \).

Step 2: Change the variable for limits

Let \(u = \frac{1}{x}\), then as \(x \to 0^+\), \(u \to +\infty\). We can rewrite \(xf\left(\frac{1}{x}\right)\) as \(\frac{1}{u} f(u)\). Thus, our task is to find \(\lim_{u \to \infty} \frac{f(u)}{u}\).

Step 3: Understand the expression for limit

Consider \(\lim_{x \to \infty} \frac{f(x)}{x}\). We know \(f'(x) = 7 - \frac{3}{4} \frac{f(x)}{x}\). If we set \(\frac{f(x)}{x} = y(x)\), we then have \(f(x) = x y(x)\).

Step 4: Substitute into the derivative

If \(f(x) = xy(x)\), then \(f'(x) = y(x) + xy'(x)\). Substituting into the differential equation: \[7 - \frac{3}{4}y(x) = y(x) + xy'(x)\], simplifying, it becomes \[xy'(x) = 7 - \frac{7}{4}y(x)\].

Step 5: Determine behavior of \(y(x)\)

The differential equation \(xy'(x) = 7 - \frac{7}{4}y(x)\) suggests a particular solution, considering constant solutions first. Set \(xy'(x) = 0\), which implies \(y(x) = \frac{4}{7}\).

Step 6: Conclusion on limit

From the calculations, as \(x \to \infty\), \(y(x) = \frac{f(x)}{x} \to \frac{4}{7}\). Therefore, \(\lim_{u \to \infty} \frac{f(u)}{u} = \frac{4}{7}\).

Step 7: Find \(\lim_{x \to 0^+} x f\left(\frac{1}{x}\right)\)

Returning to the limit, we have determined that \(\lim_{u \to \infty} \frac{f(u)}{u} = \frac{4}{7}\). Therefore, \(\lim_{x \to 0^+} x f\left(\frac{1}{x}\right) = \frac{4}{7}\).

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They describe how a particular quantity changes with respect to another, often time. Understanding differential equations is crucial because they help model real-world phenomena such as the motion of planets, growth of populations, and electrical circuits.

In this exercise, we are given a differential equation: \( f'(x) = 7 - \frac{3}{4} \frac{f(x)}{x} \). This equation ties the derivative \( f'(x) \) to the original function \( f(x) \) with respect to \( x \).

The goal is to find a solution \( f(x) \) that satisfies this equation. This requires knowing initial conditions or additional properties of \( f(x) \), such as being differentiable. By analyzing such equations, we can derive specific values or behaviors of the function, like in this exercise, where we calculate a particular limit involving \( f(x) \).

Limits

Limits are a fundamental concept in calculus and are used to describe the behavior of a function as the input approaches a certain value. They are crucial in defining continuity, derivatives, and integrals.

In this problem, we investigate the limit \( \lim _{x \rightarrow 0^{+}} x f\left(\frac{1}{x}\right) \). By changing variables, let \( u = \frac{1}{x} \), transforming the limit into \( \lim_{u \to \infty} \frac{f(u)}{u} \).
The core idea here is to see how the behavior of \( f(x) \) as \( x \to \infty \) impacts \( x f\left(\frac{1}{x}\right) \) as \( x \to 0^+ \).

• If \( f(u) \) behaves in a regular manner as \( u \) grows, the limit results in a particular constant.
• Finding that constant involves analyzing the asymptotic behavior of \( f(u) \), which in this case simplifies to \( \frac{4}{7} \).

This calculation is rooted deeply in analyzing the structure of functions and their limits when approaching infinity or zero.

Differentiable Functions

A differentiable function is one that has a derivative at every point in its domain. This indicates that the function has a well-defined tangent at every point and is smooth without any breaks or sharp edges.

The differentiability of the function \( f(x) \) in this exercise ensures that \( f(x) \) and its derivative \( f'(x) \) interact seamlessly, as given by the differential equation. A smooth, continuously changing \( f(x) \) allows us to consider its behavior over various limits and transformations.

In our context, knowing that \( f(x) \) is differentiable plays a crucial role in examining the modified function \( x f\left(\frac{1}{x}\right) \). It permits:

• Substitutions like \( f(x) = xy(x) \), leading to simplified forms of analyzed limits.
• The benefit of derivative equations, such as \( xy'(x) = 7 - \frac{7}{4}y(x) \), by assuming \( y(x) \)'s differentiable nature.

These aspects make it easier to find exact values like \( \frac{4}{7} \) or identify the function's consistent behaviors across domains.