Problem 96 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The range of values of \(a\) so that the equation \(x^{3}-3 x+\) \(a=0\) has three real and distinct roots is (A) \((-\infty,-2) \cup(2, \infty)\) (B) \((-2,0)\) (C) \((-2,0)\) (D) \((-2,2)\)

Step-by-Step Solution

Answer

The correct range for \(a\) is \((-2, 2)\), so the answer is (D).

Step 1: Understand the problem

We are given a cubic polynomial equation \(x^3 - 3x + a = 0\) and need to find the range of values for \(a\) that allow the equation to have three distinct real roots.

Step 2: Analyze roots condition for cubic polynomials

For a cubic function \(f(x)=x^3 - 3x + a\) to have three real and distinct roots, its derivative \(f'(x)\) must have two distinct real roots, such that the cubic polynomial changes sign between these two roots.

Step 3: Find critical points using the derivative

Compute the derivative: \(f'(x) = 3x^2 - 3\). Set the derivative to zero to find critical points: \(3x^2 - 3 = 0\), which simplifies to \(x^2 = 1\), so \(x = \pm 1\).

Step 4: Analyze behavior of the polynomial at critical points

Evaluate the original polynomial at the critical points: \(f(1) = 1^3 - 3\cdot1 + a = -2 + a\) and \(f(-1) = (-1)^3 - 3(-1) + a = -1 + 3 + a = 2 + a\). For three distinct roots, these must have opposite signs.

Step 5: Determine the condition for opposite signs

For \(f(1)\) and \(f(-1)\) to have opposite signs: \((-2 + a)(2 + a) < 0\). This inequality will help us find the suitable range for \(a\).

Step 6: Solve the inequality

The inequality \((-2 + a)(2 + a) < 0\) implies that \(a\) lies between the numbers \(-2\) and \(2\), excluding the endpoints. Solving, we find that \(-2 < a < 2\).

Step 7: Select the correct answer

The correct answer includes the range found in the previous step. Therefore, the answer is option (D) \((-2, 2)\).

Key Concepts

Real and Distinct Roots

To understand how a cubic polynomial can have three real and distinct roots, we must remember that each root corresponds to an x-value where the graph of the polynomial intersects the x-axis. For the equation \( x^3 - 3x + a = 0 \), having three real and distinct roots means the curve intersects the x-axis exactly three times, at different points.

When dealing with cubic polynomials, or polynomials of degree three, the general shape of the curve can vary depending on the coefficients and the constant term. However, for three real and distinct roots, the curve must change direction twice (forming two "humps") so it can cross the x-axis three times. This behavior is characterized by possessing two distinct critical points where the derivative changes sign, which we will discuss shortly in more depth.

Derivative

The derivative of a function helps us understand the function's rate of change at any given point. For the cubic polynomial \( f(x) = x^3 - 3x + a \), the derivative is calculated as \( f'(x) = 3x^2 - 3 \). This derivative is a quadratic function, which allows us to find critical points by setting it equal to zero.

Simplifying \( f'(x) = 0 \) gives \( 3x^2 - 3 = 0 \), which further reduces to \( x^2 = 1 \). Solving for x, we get the critical points: \( x = 1 \) and \( x = -1 \). These critical points are crucial because a change in the sign of the derivative around these points indicates a change in the direction of the curve's slope. This change is vital for identifying regions where the cubic can cross the x-axis a total of three times.

Critical Points

Critical points occur where the derivative of a function is zero. These points indicate where a function's slope is flat or where the curve could change direction. For \( f(x) = x^3 - 3x + a \), the critical points are \( x = 1 \) and \( x = -1 \). At these numbers, the slope of the cubic polynomial is zero, signaling possible maximum or minimum turning points.

Evaluating the polynomial at these two critical points helps us check the behavior of the function around them.

• At \( x = 1 \), \( f(1) = -2 + a \).
• At \( x = -1 \), \( f(-1) = 2 + a \).

For the polynomial to have three real and distinct roots, these values must be of opposite signs. This condition highlights the importance of critical points as they guide the transition of the curve, ensuring it crosses the x-axis in the needed locations.

Sign Change

A sign change in the context of polynomials concerns when a polynomial or function transitions from positive to negative values or vice-versa. This change indicates the function crosses the x-axis. For our cubic polynomial \( x^3 - 3x + a = 0 \), it's essential for having three distinct roots.

At the critical points \( x = 1 \) and \( x = -1 \), we have:

• \( f(1) = -2 + a \)
• \( f(-1) = 2 + a \)

To ensure opposite signs, these expressions must satisfy \((-2 + a)(2 + a) < 0\). Solving this inequality, we find that \(-2 < a < 2\). When \(a\) lies within this range, one function value will be positive and the other negative, confirming a sign change, and thus the cubic function actually crosses the x-axis three times. This condition ensures we have the desired three real and distinct roots, completing our understanding of the polynomial's behavior.