Problem 55 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Given \(P(x)=x^{4}+a x^{3}+b x^{2}+c x+d\) such that \(x=0\) is the only real root of \(P^{\prime}(x)=0\). If \(P(-1)

Step-by-Step Solution

Answer

(A) \(P(-1)\) is the minimum and \(P(1)\) is the maximum of \(P\).

Step 1: Analyze the Condition on Derivative

We are given that the only real root of \(P'(x)\) is at \(x = 0\). This indicates that \(P'(x)\) is a polynomial with a double root at \(x = 0\) and no other real roots. This means \(P(x)\) has a point of inflection at \(x = 0\).

Step 2: Evaluate Behavior of Polynomial Derivative

Since \(x = 0\) is the only real root, \(P'(x)\) does not change sign at \(x = 0\). This suggests \(P(x)\)'s slope could be either non-negative or non-positive everywhere. Let's explore the nature by taking points into consideration: \(P'(x)\) should be positive for \(x > 0\) or negative for \(x < 0\).

Step 3: Evaluate the Condition \(P(-1)<P(1)\)

Given \(P(-1) < P(1)\), this implies the function value at \(x = 1\) is greater than at \(x = -1\). This could suggest \(P(x)\) increases over \([-1, 1]\).

Step 4: Consider Extrema Possibilities

Since \(x = 0\) is a point of inflection and \(P(-1) < P(1)\), \(P(x)\) likely does not have a local maximum or minimum at \(x = 0\). Thus, inside \([-1, 1]\), \(P(-1)\) could be the minimum and \(P(1)\) the maximum.

Step 5: Determine Order of Extremes

Given \(P(-1) < P(1)\) and nature from previous steps, suggest potential ordering: \(P(-1)\) being less than \(P(1)\) further supports \(P(-1)\) as the minimum and \(P(1)\) as the maximum over \([-1, 1]\).

Key Concepts

Roots of Polynomial

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. When solving for the roots of a polynomial, we often use techniques like factoring, synthetic division, or the quadratic formula. However, the nature and number of roots can provide important information about the function's behavior.

For example, in this exercise, the derivative of the polynomial, denoted as \( P'(x) \), has a crucial role. Since it's given that \( x = 0 \) is the only real root of \( P'(x) \), this tells us that the slope of the polynomial function \( P(x) \) doesn't change sign on either side of \( x = 0 \). This point is a double root, indicating it's neither a clear maximum nor a minimum, but rather a point of inflection where the concavity of the function may change.

Understanding the behavior at the roots of the derivative, such as \( x=0 \) in this case, helps us further analyze how the main polynomial function behaves over the given intervals.

Derivatives of Polynomial

The derivative of a polynomial provides the rate at which the function changes and helps us investigate the slope of the tangent line at any point along the curve. For a given polynomial function \( P(x) \), the derivative \( P'(x) \) is calculated by applying the power rule to each term.

In this context, \( P'(x) \) is crucial for analyzing and determining the behavior of \( P(x) \). Since \( x=0 \) is the only real root of \( P'(x) \), it signifies a point where the slope of the tangent line to the graph of \( P(x) \) is zero, but the direction of the slope does not reverse. This means the polynomial does not have additional real roots in the derivative, and thus does not experience a local minimum or maximum at this point.

This derivative information sheds light on how \( P(x) \) might be increasing or decreasing over an interval, which is essential for determining possible extrema. In simpler terms, knowing how the slope behaves at these points can reveal where \( P(x) \) might reach its highest and lowest values within a given range.

Polynomial Extrema

Extrema of a polynomial refer to the maximum and minimum values that the function can attain. To find these extrema, we often analyze the function and its derivative, looking for critical points where the derivative is zero or undefined. These points help in determining where the curve reaches its highest or lowest points.

In this exercise, after analyzing \( P(x) \), it's deemed that \( x=0 \) is a point of inflection due to the characteristics of \( P'(x) \). Therefore, it doesn't represent a local maximum or minimum. Since it is given that \( P(-1)
Hence, \( P(-1) \) potentially represents the minimum value and \( P(1) \) the maximum value over the interval \([-1, 1]\). Such analysis is essential in understanding the breadth of a function's graph and verifying hypotheses about its extrema in problem-solving scenarios.