Problem 43 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f_{j}=\sum_{i=0}^{2} a_{i j} x^{i}, j=1,2,3\) and if \(f_{j}^{\prime} f_{j}^{\prime \prime}\) denote \(\frac{d f_{j}}{d x}, \frac{d^{2} f_{j}}{d x^{2}}\) respectively, then \(g(x)=\left|\begin{array}{lll}f_{1} & f_{2} & f_{3} \\\ f_{1}^{\prime} & f_{2}^{\prime} & f_{3}^{\prime} \\ f_{1}^{\prime \prime} & f_{2}^{\prime \prime} & f_{3}^{\prime \prime}\end{array}\right|\) is (A) a cubic in \(x\) (B) a quadratic in \(x\) (C) linear in \(x\) (D) a constant

Step-by-Step Solution

Answer

(A) g(x) is a cubic in x.

Step 1: Identify the Functions

We are given \( f_j = \sum_{i=0}^{2} a_{ij} x^{i} \) for \( j = 1, 2, 3 \). This means each \( f_j \) is a quadratic function of the form \( f_j = a_{0j} + a_{1j}x + a_{2j}x^2 \). We have three such functions: \( f_1, f_2, f_3 \).

Step 2: Differentiate the Functions

We find the first derivative \( f_j' = \frac{d f_j}{dx} = a_{1j} + 2a_{2j}x \) and the second derivative \( f_j'' = \frac{d^2 f_j}{dx^2} = 2a_{2j} \) for each \( j = 1, 2, 3 \).

Step 3: Set Up the Determinant

The function \( g(x) \) is defined as \[ g(x) = \left| \begin{array}{ccc} f_1 & f_2 & f_3 \ f_1' & f_2' & f_3' \ f_1'' & f_2'' & f_3'' \end{array} \right| \]. This is a 3x3 determinant with quadratic polynomials in the first row, linear polynomials in the second row, and constants in the third row.

Step 4: Expanding the Determinant

We expand the determinant along the first row. Each element of the first row is a quadratic expression multiplied by the corresponding 2x2 determinant formed from elements of the second and third rows. This effectively results in terms of degree 3 as the highest degree, establishing that result from combining a quadratic in \( x \) with linear terms.

Step 5: Conclusion on the Nature of g(x)

Since the determinant involves multiplying quadratic expressions with linear expressions, the highest degree term in the expansion will be \( x^3 \). Therefore, \( g(x) \) is a polynomial of degree 3.

Key Concepts

Derivative of polynomials

The derivative of a polynomial is a fundamental concept in calculus that helps us understand how functions change. If you have a polynomial, such as

• \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \),

we use the derivative to find the rate of change of the function. This is done by applying the power rule, which states that

• if \( f(x) = x^n \), then the derivative \( f'(x) = n x^{n-1} \).

For a polynomial, this rule is applied to each term individually, allowing you to convert higher-degree terms into lower-degree terms.
For example, if \( f(x) = 3x^2 + 2x + 1 \), the derivative would be

• \( f'(x) = 6x + 2 \).

The importance of derivatives lies in their ability to show how fast a quantity is changing at any point, which is crucial for many applications.

Second derivative

The second derivative of a function provides even deeper insights into the behavior of polynomials. Given a function \( f(x) \), the second derivative, denoted \( f''(x) \), tells us about the concavity or the "curvature" of the function.
Here's how it works:

• The first derivative \( f'(x) \) shows the slope of \( f(x) \), while the second derivative \( f''(x) \) shows how this slope changes.
• If \( f''(x) > 0 \), the function is concave up (looks like a smile), meaning it is curving upwards.
• Conversely, if \( f''(x) < 0 \), the function is concave down (frown), curving downwards.

For a quadratic function, such as \( ax^2 + bx + c \), the second derivative is simply

• \( 2a \).

Thus, for quadratic functions, the second derivative is constant, which means their concavity does not change as you move along the graph.

Polynomial degree

Understanding the degree of a polynomial is vital in recognizing the behavior and characteristics of the function. The degree of a polynomial is defined as the highest power of the variable in its expression.

• For instance, in \( 4x^3 + 3x^2 + 2 \), the highest power is 3, making it a third-degree polynomial.

The degree informs us of the maximum number of roots (or solutions) the polynomial can have. It also predicts the polynomial's end behavior as \( x \) approaches positive or negative infinity.

• In our earlier example, as \( x \to \infty \) or \( x \to -\infty \), the graph behaves like \( x^3 \), either rising or falling without bounds.

The exercise further illustrates the importance of the degree when dealing with determinants of polynomials. By understanding that combining polynomials of varying degrees results in a polynomial whose highest degree is the sum of their degrees, such insights can resolve questions regarding the nature and classification of polynomial expressions, as shown with \( g(x) \) being a cubic polynomial in the exercise.