Problem 14 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f(x)\) is a polynomial of degree \(n(>2)\) and \(f(x)=\) \(f(k-x)\), (where \(k\) is a fixed real number), then degree of \(f^{\prime}(x)\) is (A) \(n\) (B) \(n-1\) (C) \(n-2\) (D) None of these

Step-by-Step Solution

Answer

The degree of \(f'(x)\) is \(n-1\).

Step 1: Understand the Symmetry

Given the condition \(f(x) = f(k-x)\), this indicates that \(f(x)\) is symmetric about the vertical line \(x = \frac{k}{2}\). In polynomial terms, this suggests that \(f(x)\) can be written as a polynomial in \((x - \frac{k}{2})\). Specifically, given the symmetry of \(f(x)\), the terms of odd powers about this midpoint will cancel, implying that \(f(x)\) consists of even powers of \((x - \frac{k}{2})\).

Step 2: Degree Analysis of f(x)

Since \(f(x)\) is a polynomial of degree \(n\) and is symmetric about \(x = \frac{k}{2}\), it must be expressible in terms of even powers. Thus, its effective terms are of even degree. However, this symmetry itself does not reduce the actual degree \(n\), because the symmetry only implies a combination in the polynomial that appears as \(n\) with even powers centered symmetrically.

Step 3: Differentiate f(x) to find f'(x)

Differentiate the polynomial \(f(x)\) to find \(f'(x)\). The degree of \(f'(x)\) will be \(n-1\), regardless of the symmetry because differentiation will lower the power of the highest order term by one. The polynomials that are symmetric maintain their highest degree but do not change the fundamental nature of differentiation.

Step 4: Conclusion

Because differentiating a polynomial decreases its degree by 1, the degree of \(f'(x)\) is \(n-1\). The symmetry affects the form of the polynomial but does not alter the fundamental process of differentiation which reduces the degree by precisely one regardless of symmetry.

Key Concepts

Symmetry in Polynomials

Symmetry in polynomials is an intriguing aspect where the function exhibits identical values at specific intervals. If a polynomial function \( f(x) \) is symmetric around a vertical line, say \( x = \frac{k}{2} \), it means that the polynomial maintains its shape relative to this line. This symmetry implies that:

• The polynomial can be expressed in terms of even powers with respect to the line of symmetry, \((x - \frac{k}{2})\).
• Odd-power terms cancel out to maintain the symmetry when mirrored around x = k/2.

Understanding symmetry in polynomials allows for simplification in analysis and can reveal deeper properties about the function's behavior which might not be evident at first glance. Essentially, the symmetric property helps in reshaping a complex polynomial for easier manipulation.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable present in the polynomial function. For a polynomial \( f(x) \) of degree \( n \), the general form will include highest terms like \( x^n \). Here are a few essential points to note:

• The degree provides crucial information about the function's potential behavior and graph characteristics.
• For symmetric polynomials, the overall degree does not change due to symmetry.
• The maximal power denotes the leading term, which is critical in determining how many real roots a polynomial function can have.

In the given context, the degree of the polynomial, \( n \), is the starting point for understanding further operations such as differentiation, where the degree is minimally influenced by the function's inherent symmetry.

Differentiation Rules

Differentiation is a fundamental process to compute the rate at which a function changes. When differentiating a polynomial, specific rules help us find \( f'(x) \), the derivative. Here are some key rules:

• Differentiating a term \( ax^n \) results in \( anx^{n-1} \). Each term's power decreases by one while its coefficient multiplies by the original exponent.
• For the entire polynomial, polynomial differentiation reduces its degree by one.

In the context of our problem, differentiating the polynomial \( f(x) \) decreases its degree from \( n \) to \( n-1 \), irrespective of the symmetry. Although symmetry hints at specific term structures, it doesn’t alter the differentiation rule, which is consistent for all polynomial forms.