Problem 82 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(p(x)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}\) and \(|p(x)| \leq\left|e^{x-1}-1\right|\) for all \(x \geq 0\), then \(\left|a_{1}+2 a_{2}+3 a_{3}+\ldots+n a_{n}\right|\) (A) \(\leq 1\) (B) \(\geq 1\) (C) \(\geq 0\) (D) \(\leq 0\)

Step-by-Step Solution

Answer

(A) \(\leq 1\)

Step 1: Analyzing the Given Inequality

We are given the inequality \(|p(x)| \leq |e^{x-1} - 1|\) for all \(x \geq 0\). The function \(p(x)\) is a polynomial of degree \(n\), and \(e^{x-1}-1\) is the Taylor series expansion of \(e^x - e\) about \(x=1\).

Step 2: Explore Function Behavior at x=0

Evaluate both sides of the inequality at \(x=0\). We have \(p(0) = a_0\) and \(e^{0-1} - 1 = \frac{1}{e} - 1 < 0\). Thus, \(|a_0| \leq |\frac{1}{e} - 1|\).

Step 3: Consider Higher Derivatives

To find other coefficients, consider the derivatives of \(p(x)\) at \(x=0\). Specifically, evaluate \(p'(0), p''(0), \ldots\) where \(p'(x) = a_1 + 2a_2x + 3a_3x^2 + \ldots + na_nx^{n-1}\).

Step 4: Evaluate First Derivative

The first derivative at \(x=0\) gives \(p'(0) = a_1\). Given the comparison with the exponential series, the first derivative at \(x=0\) must also fall within the bounds of the derivative of \(\left|e^{x-1} - 1\right|\) chain rule applied.

Step 5: Interpretation for General Coefficient Sum

All higher coefficients coefficients in the form \(a_1 + 2a_2 + 3a_3 + \ldots + na_n\) relate to the derivatives at zero, resulting in \(|a_1 + 2a_2 + 3a_3 + \ldots + na_n| \leq |1|\) as derived from the exponential bounds.

Key Concepts

Taylor Series Expansion

Taylor series expansion is a powerful mathematical tool used to approximate functions with infinite polynomials. It allows us to express a function as an infinite sum of terms calculated from the function's derivatives at a single point.
The general formula for the Taylor series of a function \( f(x) \) centered at \( a \) is:

• \( f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \)

In the context of the original exercise, the given inequality involves the exponential function \( e^x \), which can be expressed using its Taylor series:

• \( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \)

The expression \( e^{x-1} - 1 \) can thus be expanded around the point \( x=1 \):

• \( x - 1 + \frac{(x-1)^2}{2!} + \frac{(x-1)^3}{3!} + \ldots \)

This series is crucial for understanding the bounds of the polynomial inequality provided in the exercise. It helps us compare the polynomial \( p(x) \) with the exponential function's behavior.

Higher Derivatives

In calculus, the concept of derivatives helps us understand how a function changes. While the first derivative tells us about the rate of change, higher derivatives give us additional information about the function's curvature and behavior.
For a polynomial \( p(x) \), these higher derivatives are crucial in finding the polynomial coefficients.

• \( p'(x) = a_1 + 2a_2x + 3a_3x^2 + \ldots \)
• \( p''(x) = 2a_2 + 6a_3x + \ldots \)

In the original problem, evaluating these derivatives at \( x=0 \) is essential to assess the higher derivatives of the polynomial and relate them to the exponential function it is being compared with. Each derivative helps track how the series of the polynomial matches the series of the exponential function in terms of magnitude and behavior.
The focus on \( x=0 \) stems from the Taylor expansion's reliance on a single center point, making it simpler to grasp the polynomial’s behavior at that specific point.

Exponential Function

The exponential function, \( e^x \), is fundamental in mathematics due to its unique properties and numerous applications, especially in differential equations and calculus.
Its defining quality is that its derivative is still \( e^x \), reflecting an inherent constant growth rate. This self-replicating characteristic makes it ideal for modeling naturally growing processes.

• Exponential growth or decay is natural, such as population growth or radioactive decay.
• It showcases continuous compound interest phenomena in finance.

In the context of the given inequality \( |p(x)| \leq |e^{x-1} - 1| \), the exponential function's rapid convergence through Taylor series provides a robust framework for establishing bounds on the polynomial \( p(x) \).
This constraint ensures that the variance of \( p(x) \) remains tethered to a known standard — the perfectly predictable form of \( e^x \). Consequently, controlling \( p(x) \) is inherently linked to understanding \( e^x \)’s series expansion and embodies the polynomial's maximum allowable deviation within certain parameters, serving as an upper boundary condition for the task.