Problem 60 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f(x)\) be a polynomial function of second degree. If \(f(1)=f(-1)\) and \(a_{1}, a_{2}, a_{3}\) are in A. P., then \(f^{\prime}\left(a_{1}\right)\), \(f^{\prime}\left(a_{2}\right), f^{\prime}\left(a_{3}\right)\) are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

Step-by-Step Solution

Answer

(A) A.P.

Step 1: Identify Polynomial Form and Condition

Consider the polynomial function of second degree: \( f(x) = ax^2 + bx + c \). The given condition \( f(1) = f(-1) \) indicates symmetry about the y-axis, which leads to \( b = 0 \), simplifying the polynomial to \( f(x) = ax^2 + c \).

Step 2: Define Terms of A.P. for Roots

Since \(a_1, a_2, a_3\) are in Arithmetic Progression (A.P.), we assume \( a_2 = a_1 + d \) and \( a_3 = a_1 + 2d \).

Step 3: Compute Derivative of Polynomial

Calculate the derivative of \( f(x) = ax^2 + c \). The derivative is \( f'(x) = 2ax \).

Step 4: Find Derivatives at A.P. Terms

Determine \( f'(a_1), f'(a_2), \) and \( f'(a_3) \): \[ f'(a_1) = 2aa_1, \quad f'(a_2) = 2a(a_1 + d) = 2aa_1 + 2ad, \quad f'(a_3) = 2a(a_1 + 2d) = 2aa_1 + 4ad \].

Step 5: Analyze Derivative Values for A.P., G.P., and H.P.

Observe the sequence \( f'(a_1), f'(a_2), \) and \( f'(a_3) \): \[ 2aa_1, \, 2aa_1 + 2ad, \, 2aa_1 + 4ad \].The difference between consecutive terms is constant: \[ (2aa_1 + 2ad) - 2aa_1 = 2ad, \quad (2aa_1 + 4ad) - (2aa_1 + 2ad) = 2ad \]. Since the difference is constant, the sequence is an A.P.

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence of numbers in which the difference between consecutive terms is constant. This consistent difference is known as the "common difference." For example, the sequence 2, 5, 8, 11 is an A.P. with a common difference of 3. In the context of polynomial functions, if we know that values such as roots or other critical points are in arithmetic progression, it means:

• The numbers are evenly spaced along the domain.
• For any three numbers, the middle one is the average of the other two.
• This can aid in calculations such as testing properties like symmetry or consistency within sequences.

In the exercise provided, the sequence of terms given by the derivatives of the polynomial at specific points forms an A.P. This is proven by computing the derivatives and showing the differences between consecutive derivatives remain constant. Understanding A.P. helps in recognizing patterns and simplifications in mathematical problems.

Derivatives of Polynomials

A derivative measures how a function changes as its input changes. For polynomial functions, derivatives are particularly straightforward to calculate. If you have a polynomial like \(f(x) = ax^2 + bx + c\), applying the power rule gives us the derivative. For this specific case, the derivative would be \(f'(x) = 2ax + b\). The power rule involves bringing down the exponent as a multiplier and reducing the exponent by one.In our exercise, the polynomial simplifies to \(f(x) = ax^2 + c\). Therefore, the derivative becomes \(f'(x) = 2ax\) because the linear term \(b\) is eliminated due to symmetry. Calculating derivatives provides us with a powerful tool to identify function behavior - like where it might have a maximum or minimum, where it slopes up or down, or in this case, to connect changes at different points that follow a specific sequence, such as an arithmetic progression.

Symmetry in Functions

Symmetry in functions means that the function exhibits a balanced and consistent pattern around a central point or axis. For polynomial functions, symmetry can make them much easier to handle and analyze. In the exercise, the condition given, \(f(1) = f(-1)\), implies the function is symmetric about the y-axis.

• This results because the even-powered terms remain unchanged when \(x\) is replaced with \(-x\), while odd-powered terms change their sign.
• Consequently, for a polynomial à la \(f(x) = ax^2 + bx + c\), if the function is symmetric about the y-axis, the coefficient \(b\) of the linear term must be zero, which simplifies the polynomial significantly.

Identifying symmetry can be a shortcut to simplifying problems; it helps zero in on critical terms or conditions that affect the entire function uniformly, such as reducing a polynomial function of second degree to only one involving powers of x to the second degree or constant terms.