Problem 74 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(y=a \log |x|+b x^{2}+x\) has its extremum values at \(x=-1\) and \(x=2\), then (A) \(a=2, b=-1\) (B) \(a=2, b=-1 / 2\) (C) \(a=-2, b=1 / 2\) (D) None of these

Step-by-Step Solution

Answer

The answer is (D) None of these.

Step 1: Find the derivative

The given function is \( y = a \log |x| + b x^2 + x \). To find the extremum values, we first need to find its derivative. Using the rules of differentiation, we have:\[y' = \frac{d}{dx}(a \log |x|) + \frac{d}{dx}(bx^2) + \frac{d}{dx}(x)\]\[y' = \frac{a}{x} + 2bx + 1\]

Step 2: Set the derivative to zero at extremum points

The function has extremum points at \( x = -1 \) and \( x = 2 \). We set the derivative, \( y' = \frac{a}{x} + 2bx + 1 \), equal to zero at these points:For \( x = -1 \), \( \frac{a}{-1} + 2b(-1) + 1 = 0 \) \(-a - 2b + 1 = 0 \) Equation 1: \(-a - 2b = -1 \)For \( x = 2 \), \( \frac{a}{2} + 2b(2) + 1 = 0 \) \(\frac{a}{2} + 4b + 1 = 0 \) Equation 2: \(\frac{a}{2} + 4b = -1 \)

Step 3: Solve the system of equations

We have the following system of equations from Step 2:1. \(-a - 2b = -1 \)2. \(\frac{a}{2} + 4b = -1 \)Let's solve these two equations simultaneously.First, solve equation 1 for \(a\):\( a = -1 - 2b \).Substitute \(a\) in equation 2:\[ \frac{-1 - 2b}{2} + 4b = -1 \]Multiply through by 2 to clear the fraction:\[-1 - 2b + 8b = -2 \]Combine like terms:\[ 6b = -1 \]Divide by 6:\[ b = -\frac{1}{2} \]

Step 4: Find the value of \(a\)

Using \( b = -\frac{1}{2} \) from Step 3 in equation 1:\[ a = -1 - 2(-\frac{1}{2}) \]Simplify:\[ a = -1 + 1 = 0 \]

Step 5: Verify the solution with options

From Step 4, we have \( a = 0 \) and \( b = -\frac{1}{2} \). None of the options \((A) \ a=2, b=-1\), \((B) \ a=2, b=-\frac{1}{2}\), \((C) \ a=-2, b=\frac{1}{2}\) match these values.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that deals with how a function changes as its inputs change. In simpler terms, it measures the rate at which something changes. When differentiating a function, we obtain another function called the derivative, which provides the slope of the tangent line at any given point on the curve.

In our example, we have the function \( y = a \log |x| + b x^2 + x \). To find where this function reaches extremum values (maximum or minimum points), we must first find its derivative. This is because extremum points occur where the slope of the function is zero, which is where the derivative is zero.

Using differentiation rules, the derivative \( y' \) of our function is \( \frac{a}{x} + 2bx + 1 \). We found this by applying the power rule and the logarithmic differentiation rule, which specifically helps us find the derivative of \( \log |x| \). Once we have \( y' \), the task is then to analyze where this expression equals zero, as these will define our extremum points.

System of Equations

A system of equations is a collection of two or more equations with a common set of unknowns. Solving these equations simultaneously allows us to find the specific values of the variables that satisfy all equations in the system.

In our problem, after finding the derivative and setting it to zero at given extremum points, we formed two equations: \(-a - 2b = -1\) and \(\frac{a}{2} + 4b = -1\). These two equations create a system that we need to solve to find the values of \(a\) and \(b\).

We approach these by first isolating one of the variables, in this case \(a\), from one of the equations. Then, substituting this expression for \(a\) into the other equation allows us to solve for \(b\). Once \(b\) is known, we can substitute it back to find \(a\). This process of substitution is common practice for solving simple linear systems, especially when they involve two equations.

Logarithmic Function

Logarithmic functions are the inverse of exponential functions and are fundamental in various applications of mathematics, including differentiation. A logarithmic function with the natural logarithm is often seen because it simplifies complex calculations with its simple differentiation rule.

In the given function \( y = a \log |x| + b x^2 + x \), the term \( a \log |x| \) represents the logarithmic part. The natural logarithm \( \log |x| \) is typically used because its derivative is straightforward: \( \frac{d}{dx}(\log |x|) = \frac{1}{x} \), as long as \( x eq 0 \). This derivative plays a crucial role in further simplifying the derivative of the entire function.

Logarithms help in modeling various real-world scenarios where growth diminishes over time, such as in finance or population studies. Understanding the role of logarithmic functions in such models is crucial for solving different mathematical problems, including the one we're dealing with here.