Problem 87 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation \(x^{5}-3 x-1=0\) has, in the interval \([1,2]\) (A) at least one root (B) at most one root (C) no root (D) a unique root

Step-by-Step Solution

Answer

The equation has a unique root in the interval \([1, 2]\) (Option D).

Step 1: Understand the Problem

We need to determine if the equation \(x^{5} - 3x - 1 = 0\) has roots in the interval \([1, 2]\). We will consider if there is at least one root, no root, or a unique root within this interval.

Step 2: Evaluate the Function at the Endpoints

Start by evaluating the function \(f(x) = x^5 - 3x - 1\) at the endpoints of the interval. Compute:- \(f(1) = 1^5 - 3 \times 1 - 1 = -3\)- \(f(2) = 2^5 - 3 \times 2 - 1 = 24 - 6 - 1 = 17\).

Step 3: Apply the Intermediate Value Theorem

The Intermediate Value Theorem states that if \(f(x)\) is continuous on \([1, 2]\) and \(f(1)\) and \(f(2)\) have opposite signs, then there is at least one root in \((1, 2)\). Here, \(f(1) = -3\) and \(f(2) = 17\). The function is continuous and changes sign, so there is at least one root.

Step 4: Determine If There is More Than One Root

To check if there is exactly one root, analyze the derivative \(f'(x) = 5x^4 - 3\). On the interval \([1, 2]\), \(f'(x)\) will always be positive, meaning \(f(x)\) is strictly increasing. Since the function is strictly increasing and there is a sign change, there must be exactly one root.

Key Concepts

Roots of Polynomials

When thinking about the roots of polynomials, you are looking for values of the variable that make the polynomial equal to zero. Finding these values, known as roots, or solutions, is crucial in many areas of mathematics. In our problem, you need to find where the polynomial equation \(x^5 - 3x - 1 = 0\) equals zero on the interval \([1, 2]\). If you plot the polynomial, roots appear where the graph intersects the x-axis.
Below are some clear steps to help understand roots of polynomials:

• Evaluate the polynomial: Start by substituting values to see if they produce zero.
• Identify sign changes: Check the function at different points; a switch from positive to negative indicates a potential root.
• Estimate and verify: Use the sign change to locate a likely interval, then test points or use methods like bisection to home in on the exact root.

These steps can greatly simplify the process of finding roots.

Calculating Derivatives

Calculating derivatives is essential for understanding how a function behaves along its curve. The derivative of a polynomial function gives you the rate of change or slope of the function at any point. For solving our problem, the derivative \(f'(x) = 5x^4 - 3\) is considered. For many functions:

• Compute the derivative: Derivatives of polynomials are found using power and chain rules.
• Analyze the derivative: By studying \(f'(x)\), you discover when a function is increasing or decreasing.
• Apply the results: If \(f'(x) > 0\) over the interval, the function is increasing, suggesting one root due to the Intermediate Value Theorem.

Understanding derivatives allows for more profound insights into function characteristics, like where peaks, troughs, or roots occur.

Continuous Functions

A continuous function is one where you can draw its graph without lifting your pencil. This smoothness is vital in applying helpful mathematical theorems like the Intermediate Value Theorem, which we used in the original solution for our polynomial.In the scenario we explored:

• Determine continuity: Check if you can plot the function with no breaks in it. Polynomials, like \(x^5 - 3x - 1\), are always continuous.
• Use the Intermediate Value Theorem: If the function is continuous over the interval and there is a sign change between \(f(a)\) and \(f(b)\), then at least one root exists in that interval.

This property was crucial for establishing that a root exists between \([1, 2]\), confirming the existence and uniqueness of the root.