Problem 2 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(b^{2}-4 a c=0\) and \(a>0\), then the domain of the function \(f(x)=\log \left(a x^{3}+(2 a+b) x^{2}+(2 b+c) x+2 c\right)\) is (A) \((-2, \infty) \backslash\left\\{-\frac{b}{2 a}\right\\}\) (B) \([-2, \infty) \backslash\left\\{-\frac{b}{2 a}\right\\}\) (C) \((-\infty,-2) \backslash\left\\{-\frac{b}{2 a}\right\\}\) (D) None of these

Step-by-Step Solution

Answer

The domain is \((-2, \, \infty) \setminus \{-\frac{b}{2a}\}\). (Option A)

Step 1: Understanding the Problem

We need to find the domain of the logarithmic function \(f(x) = \log(a x^3 + (2a+b)x^2 + (2b+c)x + 2c)\). The expression inside the logarithm must be greater than zero. Additionally, given \(b^2 - 4ac = 0\), we know the quadratic equation has one unique real root.

Step 2: Substitution and Simplification

Since \(b^2 - 4ac = 0\), the quadratic equation \(ax^2 + bx + c = 0\) has a double root at \(x = -\frac{b}{2a}\). Substitute this into our expression: \(x = -\frac{b}{2a}\). Thus, the original cubic polynomial at this point becomes zero or a turning point.

Step 3: Analysis of Polynomial Behavior

Analyze the polynomial's behavior using the double root \(x = -\frac{b}{2a}\). Given this is a cubic expression due to higher powers of \(x\), test intervals around \(x = -\frac{b}{2a}\) and \(x = -2\) to determine where the expression is positive.

Step 4: Interval Testing for Domain

1. Choose test points in intervals: \((-\infty, -2)\), \((-2, -\frac{b}{2a})\), and \((-\frac{b}{2a}, \infty)\).2. Calculate the sign of the expression in these intervals; the function is defined where it is positive.3. Since \(a > 0\), test intervals suggest positivity from \(-2\) to \(\infty\), but excluding \(-\frac{b}{2a}\) due to zero at this point.

Step 5: Conclusion

Hence, the expression inside the log function is greater than zero for \(( -2, \, \infty)\), but undefined at \(-\frac{b}{2a}\), which must be excluded from the domain.

Key Concepts

Domain of a Function

When working with functions, especially in mathematics, the domain of a function is crucial. It represents all the possible input values that a function can accept. For the function \( f(x) = \log(a x^3 + (2a+b)x^2 + (2b+c)x + 2c) \), the domain is determined by the expression inside the logarithm.
This specific expression must always be positive, as logarithms of zero or negative numbers are undefined.
Thus, we must ensure that the cubic polynomial inside the log is greater than zero. This involves solving inequalities to find where the expression is positive, and checking critical points where it may turn zero.

Logarithmic Function

Logarithmic functions are vital in mathematics, especially in complex equations. They are the inverses of exponential functions. The natural logarithm, \( \log(x) \), only accepts positive argument values.
For \( f(x) = \log(a x^3 + (2a+b)x^2 + (2b+c)x + 2c) \), solely positive values of the polynomial are valid.

• Zero or negative arguments result in undefined outputs.
• Thus, finding where the polynomial is positive specifies the domain.

Understanding this property helps pinpoint which sections along the x-axis (input range) are part of the domain.

Quadratic Equations

Quadratic equations, often represented as \( ax^2 + bx + c = 0 \), play a role in this exercise. They help identify key properties of the expression inside the logarithm. Specifically, \( b^2 - 4ac = 0 \) reveals that the quadratic has one unique root: \( x = -\frac{b}{2a} \).
This unique solution means the quadratic expression is a perfect square, reaching zero only at this point. It also acts as a turning point in the polynomial behavior around this x-value. The double root property is crucial in segmenting the domain, helping to exclude points where the expression goes to zero.

Polynomial Analysis

Polynomial analysis involves studying the properties and behaviors of polynomial functions like the cubic polynomial in this task.
By breaking it into intervals, we explore where the function is positive or negative, which dictates the domain.

• Identify roots and their multiplicities.
• Test intervals between roots to see sign changes.

For the cubic \( a x^3 + (2a+b)x^2 + (2b+c)x + 2c \):
If \( a > 0 \), the graph tends to rise from left to right, impacting positivity intervals respectively.
Critical points, like \( -\frac{b}{2a} \), create transitions in sign or behavior, requiring analytical checks step-by-step.
This methodical analysis helps in determining where the logarithmic function is valid.