Problem 2 - A Complete Resource Book in Mathematics for JEE Main 2017

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) \([-2, \infty)\left\\{-\frac{b}{2 a}\right\\}\) (C) \((-\infty,-2) \backslash\left\\{-\frac{b}{2 a}\right\\}\) (D) None of these

Step-by-Step Solution

Answer

(A) \((-2, \infty) \backslash\{-\frac{b}{2a}\}\).

Step 1: Analyze the Quadratic Discriminant

Given the equation discriminant condition \(b^{2}-4ac=0\), we know this means the quadratic \(ax^2 + bx + c\) has a double root or perfect square root. Thus, the double root is \(x = -\frac{b}{2a}\).

Step 2: Function Inside Logarithm

The expression inside the logarithm, \(g(x) = ax^{3}+(2a+b)x^{2}+(2b+c)x+2c\), is a cubic polynomial. For the logarithm function to be defined, \(g(x) > 0\).

Step 3: Analyze Critical Points

From Step 1, we found that the quadratic \(ax^2 + bx + c\) has a double root at \(x = -\frac{b}{2a}\). At this root, the cubic function might change its sign or have the value where it is identically zero.

Step 4: Behavior Around x = -2

Consider testing values to the right of \(x = -2\). Check if \(g(-2)\) is positive or negative and determine the behavior of the cubic around this point. If \(g(-2) > 0\), the function is possibly positive for \(x > -2\).

Step 5: Conclusion on the Domain

Given that \(g(x)\) needs to be positive and noting the double root behavior, the domain is \((-2, \infty)\) excluding \(x = -\frac{b}{2a}\), which might cause \(g(x) = 0\). Therefore, the domain is \((-2, \infty)\setminus \{-\frac{b}{2a}\}\).

Step 6: Choose the Correct Option

The correct answer from the options given is (A), which accurately represents the domain as \((-2, \infty) \backslash\{-\frac{b}{2a}\}\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, generally written in the form \(ax^2 + bx + c = 0\). It has several important properties and behaviors, largely determined by the coefficients \(a\), \(b\), and \(c\).

The discriminant of a quadratic equation, expressed as \(b^2 - 4ac\), plays a crucial role in understanding the nature of its roots.

• If \(b^2 - 4ac > 0\), the quadratic will have two distinct real roots.
• If \(b^2 - 4ac = 0\), it will have exactly one real root, known as a double or repeated root.
• If \(b^2 - 4ac < 0\), the roots will be complex and not real.

In the exercise, the condition given \(b^2 - 4ac = 0\) indicates a perfect square trinomial, meaning our quadratic \(ax^2 + bx + c\) has a double root.

Understanding this property helps us predict the behavior of certain points, particularly where the polynomial touches the x-axis only once, which is generally at the root \(x = -\frac{b}{2a}\). This is fundamental when deciding domains for functions involving quadratic or higher-degree polynomials.

Cubic Polynomial

Cubic polynomials have the general form \(ax^3 + bx^2 + cx + d\), where the highest power of the variable is three. These polynomials can exhibit more complex behavior than quadratics because of their higher degree.

• They can have up to three real roots.
• There could be turning points, usually at a minimum of two like a crest and a trough.
• Their graph can cross the x-axis up to three times based on the number of real roots.

In our context, the function \(g(x) = ax^{3}+(2a+b)x^{2}+(2b+c)x+2c\) inside the logarithm plays a critical role since it's a cubic polynomial.

Its behavior at significant points like \(x = -\frac{b}{2a}\) and around \(-2\) must be carefully analyzed. We know from the quadratic analysis that \(x = -\frac{b}{2a}\) is a double root of a related quadratic, influencing the shape and characteristics of the cubic.

To understand the domain of the logarithmic function involving this cubic, we must ensure \(g(x)\) stays positive over specific intervals. This is vital for determining where the logarithm, demanding positive values, is defined.

Logarithmic Function

A logarithmic function is expressed as \(f(x) = \log(g(x))\), and it is only defined for positive values of \(g(x)\), as the logarithm of a non-positive number is undefined in real numbers.

It's crucial to identify intervals where \(g(x) > 0\). If \(g(x)\) becomes zero or negative, \(f(x)\) won't be defined.

• These conditions lead to constraints on its domain.
• Checking various points on the interval ensures \(g(x)\) remains positive.

For our problem, we inspect the cubic \(g(x) = ax^{3}+(2a+b)x^{2}+(2b+c)x+2c\) over an interval as decided by our quadratic discriminant analysis and further polynomial behavior.

By careful analysis, the domain of the function is identified as \((-2, \infty)\backslash\{-\frac{b}{2a}\}\), making certain the function is defined only where \(g(x) > 0\).
This understanding of the logarithmic function's dependency on its inner polynomial function guides the correct choice of domain in any similar problem.

Domain Analysis

Domain analysis involves determining all possible x-values for which a function is defined. For our function \(f(x) = \log(g(x))\) to be well-defined, \(g(x) > 0\) as discussed.

Determining the domain requires:

• Identifying where \(g(x)\) transitions from negative to positive or vice versa.
• Excluding points where \(g(x) = 0\) as these make \(f(x)\) undefined.
• Considering behavior at critical points, such as roots and intersections with axes.

In our problem, assessing from the roots and double-root behavior, we conclude the domain.

Our quadratic discriminant \(b^2 - 4ac = 0\) involved gives a transformative insight into how it impacts broader intervals – for instance, excluding \(-\frac{b}{2a}\) ensures continuity.
By complementing mathematical reasoning with value checks around \(x = -2\), we confirmed that beyond \(-2\), \(g(x)\)'s positivity is sustained without hitting zero at \(-\frac{b}{2a}\), thus yielding the chosen domain. Such analysis highlights the fusion of polynomial roots, intervals, and positivity as paramount in domain exploration.