Problem 183 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(x=-1\) and \(x=2\) are extreme points of \(f(x)=\alpha \log |x|+\beta x^{2}+x\), then (A) \(\alpha=-6, \beta=\frac{1}{2}\) (B) \(\alpha=-6, \beta=-\frac{1}{2}\) (C) \(\alpha=2, \beta=-\frac{1}{2}\) (D) \(\alpha=2, \beta=\frac{1}{2}\)

Step-by-Step Solution

Answer

Option (C): \(\alpha = 2\), \(\beta = -\frac{1}{2}\).

Step 1: Determine the derivative of the function

First, calculate the derivative of the function \( f(x) = \alpha \log |x| + \beta x^{2} + x \). The derivative is given by \( f'(x) = \frac{\alpha}{x} + 2\beta x + 1 \).

Step 2: Find the conditions for extreme points

For \( x = -1 \) and \( x = 2 \) to be extreme points, the derivative must equal zero at these points. Therefore, set \( f'(-1) = 0 \) and \( f'(2) = 0 \).

Step 3: Solve for \(\alpha\) using the first extreme point

Substitute \( x = -1 \) into the derivative and set it to zero: \( f'(-1) = \frac{\alpha}{-1} + 2\beta(-1) + 1 = 0 \). Simplify this to \( -\alpha - 2\beta + 1 = 0 \), or \( \alpha = -2\beta + 1 \).

Step 4: Solve for \(\alpha\) and \(\beta\) using the second extreme point

Substitute \( x = 2 \) into the derivative and set it to zero: \( f'(2) = \frac{\alpha}{2} + 2\beta(2) + 1 = 0 \). Simplify this to \( \frac{\alpha}{2} + 4\beta + 1 = 0 \), or \( \alpha = -8\beta - 2 \).

Step 5: Solve the system of equations for \(\alpha\) and \(\beta\)

You now have two equations: \( \alpha = -2\beta + 1 \) and \( \alpha = -8\beta - 2 \). Set them equal to each other: \( -2\beta + 1 = -8\beta - 2 \). Solve for \(\beta\): \(6\beta = -3 \rightarrow \beta = -\frac{1}{2} \).

Step 6: Calculate \(\alpha\) using \(\beta\) value

Substitute \( \beta = -\frac{1}{2} \) back into the equation for \(\alpha\): \( \alpha = -2(-\frac{1}{2}) + 1 = 1 + 1 = 2 \).

Step 7: Choose the correct option based on \(\alpha\) and \(\beta\)

You found \( \alpha = 2 \) and \( \beta = -\frac{1}{2} \), which matches option (C).

Key Concepts

Derivative

A derivative is a fundamental concept in calculus. It represents the rate at which a function is changing at any given point. In practical terms, if you think of a graph as a road, the derivative shows you the slope of that road at every point.

For example, if our function is \( f(x) = \alpha \log |x| + \beta x^{2} + x \), then its derivative, \( f'(x) \), gives us a way to find where the function increases, decreases, or stays constant. We calculate it using rules of differentiation, which may involve:

• Power Rule: \( \frac{d}{dx} x^n = nx^{n-1} \)
• Logarithmic Rule: \( \frac{d}{dx} \log |x| = \frac{1}{x} \)

The derivative of our function is \( f'(x) = \frac{\alpha}{x} + 2\beta x + 1 \). This equation allows us to explore the function's behavior and identify extreme points where the derivative equals zero.

System of Equations

A system of equations consists of multiple equations that share the same set of variables. The goal is to find the values of those variables that satisfy all the equations simultaneously.

In our problem, we derive two equations from the condition that the derivative equals zero at the extreme points \( x = -1 \) and \( x = 2 \). This gives us:

• \( \alpha = -2\beta + 1 \)
• \( \alpha = -8\beta - 2 \)

These two equations form a system that can be solved to find \( \alpha \) and \( \beta \). By setting the equations equal, a crucial step is isolating one variable, like \( \beta \), which helps us find its value. Then, substituting back to find \( \alpha \), we successfully solve the system. This technique is widespread, offering a blueprint for various mathematical and real-world applications.

Logarithmic Function

Logarithmic functions are the inverse of exponential functions and have a wide range of applications. In the function \( f(x) = \alpha \log |x| + \beta x^{2} + x \), the term \( \alpha \log |x| \) showcases a logarithmic characteristic.

Logarithmic functions increase quickly but then gradually level off as \( x \) increases. This behavior is important in describing phenomena such as sound intensity, earthquake magnitudes, or mathematical models where rapid initial growth stabilizes.

The absolute value \( |x| \) in our logarithmic term signifies how it accommodates both positive and negative \( x \) values smoothly, though it results in complex numbers for \( x = 0 \). Understanding the nature of logarithmic functions aids in conceptualizing how they transform inputs logarithmically and how such changes impact the overall function \( f(x) \).