Problem 136 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a+b+c=0\), then the equation \(3 a x^{2}+2 b x+c=0\) has, in the interval \((0,1)\) (A) at least one root (B) at most one root (C) no root (D) None of these

Step-by-Step Solution

Answer

The equation has at least one root in the interval \((0, 1)\). (A) at least one root.

Step 1: Understanding the Given Condition

We are given that the sum of the coefficients \( a + b + c = 0 \). This will be crucial in simplifying and solving the quadratic equation later.

Step 2: Rewriting the Quadratic Equation

The quadratic equation is given by \( 3ax^2 + 2bx + c = 0 \). Using \( a + b + c = 0 \), we can rewrite \( c = - (a + b) \). Substitute \( c = - (a + b) \) into the equation:\[ 3ax^2 + 2bx - (a + b) = 0 \]

Step 3: Simplifying the Equation in the Interval

Focus on the behavior of the equation in the interval \((0,1)\). We'll need to manipulate and analyze it for roots between these bounds. The property that \( a+b+c=0 \) will help simplify our assessment.

Step 4: Testing for Roots within the Interval

At the endpoints and within the interval, substitute simple values of \( x \) and check if they satisfy the equation. For example, substitute \( x = 0 \) and \( x = 1 \), but focus mainly on internal values. If such a root exists within, it will prove a root in \((0,1)\).

Step 5: Finding Parameters at Special Cases

Evaluate how parameters such as derivatives or specific test values relate to having roots in the interval \((0, 1)\). Use derivative signs or test-values strategies to handle proofs of roots existing within the interval.

Step 6: Conclusion from the Properties

Given the equation is continuous, it either has zero or an even number of roots in parts where it crosses the x-axis. For a simple, continuous polynomial, if it approaches zero towards an endpoint, it follows there may exist a root in the open interval.

Key Concepts

Roots of Equation

In the context of quadratic equations, understanding the 'roots of the equation' is crucial. Roots refer to the values of the variable, usually denoted as \( x \), that satisfy the equation, making it equal zero. For example, in the equation \( ax^2 + bx + c = 0 \), the roots are the values of \( x \) that solve this equation.

To find these roots, we either factor the quadratic expression or use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). However, in some cases, like our original exercise, additional conditions can lead us to deduce the presence or absence of roots without direct solving.

• Given a specific interval, such as \((0,1)\), we determine roots using substitution and validation within these bounds.
• Other methods may involve using specific properties of the equation, such as the sum of coefficients, as with \( a + b + c = 0 \).

Recognizing these relationships simplifies finding roots and understanding conditions under which roots exist.

Interval Testing

Interval testing is a strategy used to identify the presence of roots within a specific range or boundary. For quadratic equations, analyzing intervals helps us understand where the polynomial might cross the x-axis.

The basic idea is to test various values of \( x \) within the provided interval to see if a change in sign occurs. The actual presence of a root is indicated when the equation changes signs, say from positive to negative or vice versa, when passing through some \( x \) value:

• Begin by substituting the endpoints of the interval into the equation and evaluate the results. Note whether the results are positive or negative.
• Check for any intermediate values within the interval to see if any zero crossing occurs, indicating a root might exist.

Using this method rigorously ensures that we actually find roots within a particular part of the x-axis, giving solid ground for understanding the solution set of polynomial equations.

Polynomial Continuity

Polynomial continuity refers to the smooth, unbroken, and often predictable nature of polynomial functions. A polynomial’s continuity ensures that it has no sudden jumps or holes in its graphical representation, making it a continuous curve across the entire range of real numbers.

In the context of finding roots, continuity plays a vital role because it allows us to rely on principles such as the Intermediate Value Theorem (IVT). The theorem states:

• If a continuous function, like a polynomial, takes on different signs at two points within an interval, it must cross the x-axis somewhere in between.
• Given the polynomial is continuous over the interval \((0, 1)\), if changes in sign can be established at any point, then we know a root must exist within that interval.

Understanding continuity helps predict behavior across intervals, completing our comprehension of the roots within defined limits without needing to solve for them directly.