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

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

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

Step 1: Verify the function is continuous

First, we need to confirm that the function \( f(x) = x^5 - 3x - 1 \) is continuous on the interval \([1, 2]\). Polynomial functions like this one are continuous everywhere, so it is continuous on \([1, 2]\).

Step 2: Evaluate the function at the endpoints

To apply the Intermediate Value Theorem, evaluate \( f(x) \) at the endpoints of the interval: \(f(1) = 1^5 - 3 \times 1 - 1 = -3 \)\(f(2) = 2^5 - 3 \times 2 - 1 = 32 - 6 - 1 = 25 \)

Step 3: Apply the Intermediate Value Theorem

The function values at the endpoints are \( f(1) = -3 \) and \( f(2) = 25 \). Since \( f \) changes sign and is continuous, by the Intermediate Value Theorem, there is at least one root in the interval \([1, 2]\).

Step 4: Evaluate derivative to determine number of roots

To establish the number of roots, check if the function is strictly increasing or decreasing by evaluating its derivative:\( f'(x) = 5x^4 - 3 \).Examine \( f'(x) \) in the interval \([1, 2]\):- At \( x = 1 \), \( f'(1) = 5(1)^4 - 3 = 2 \).- At \( x = 2 \), \( f'(2) = 5(2)^4 - 3 = 77 \).Since \( f'(x) > 0 \) throughout the interval, \( f(x) \) is strictly increasing on \([1, 2]\).

Step 5: Conclude the uniqueness of the root

Since the function is strictly increasing on \([1, 2]\) and continuous, it crosses the x-axis only once. Thus, there is exactly one root in the interval.

Key Concepts

Polynomial Function

Polynomial functions are a type of mathematical expression involving sums of powers of a variable. They can take a very general form: \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants and \(n\) is a non-negative integer, representing the degree of the polynomial. In the exercise provided, our polynomial is \(f(x) = x^5 - 3x - 1\), which is a fifth-degree polynomial.

Some unique characteristics include:

• They are smooth and continuous curves.
• They can have a variety of shapes, depending on the degree and coefficients.
• The highest power allows us to determine key features like the total number of possible roots throughout the entire real number line.

Understanding polynomial functions is critical since they frequently show up in various branches of mathematics, including calculus, where they help describe more complex objects.

Continuous Function

A continuous function is one that you can draw without lifting your pencil from the paper. This means that the function has no abrupt breaks or changes in its value.

The Intermediate Value Theorem, crucial in this exercise, relies on this property of continuous functions. Because polynomial functions are continuous everywhere, the function \(f(x) = x^5 - 3x - 1\) maintains this property throughout the interval \([1, 2]\).

Continuous functions have several important aspects:

• They allow us to apply the Intermediate Value Theorem, which helps determine roots within specific intervals.
• They provide assurance that any sign change within an interval implies the existence of a root.

In our exercise, the continuity of the function \(f(x)\) over \([1, 2]\) ensures that when the function value changes from -3 to 25, there must be a point it equals zero, i.e., a root.

Derivative Analysis

Analyzing derivatives gives us a deeper insight into the behavior of functions, particularly when we're interested in behavior like increasing or decreasing trends. The derivative of a function represents its rate of change; in this case, the derivative is \(f'(x) = 5x^4 - 3\).

Evaluating this derivative allows us to examine how the function behaves across its domain, especially on specific intervals:

• Derivative analysis tells us where the function is increasing or decreasing.
• A positive derivative, as found here, indicates that the function is increasing.
• By investigating \(f'(x)\) at points in \([1, 2]\), we find that the derivative remains positive, which conveys that the function keeps increasing.

This increase in the function shows us that \(f(x)\) changes continuously without turning back on itself, which aids in confirming the uniqueness of a root.

Strictly Increasing Function

A strictly increasing function is one where for every pair of points \(x_1\) and \(x_2\), if \(x_1 < x_2\), then \(f(x_1) < f(x_2)\). It means the function never flattens or turns down as \(x\) increases. This property greatly simplifies locating and concluding the uniqueness of roots within an interval.

With \(f(x) = x^5 - 3x - 1\), we determined its derivative was positive across \([1, 2]\), confirming the function is strictly increasing in this range.

The benefits of establishing that a function is strictly increasing are:

• It implies that there can be only one crossing point or root within any range where it applies.
• It prevents the function from "doubling back", giving easy reasoning that the graph intersects the x-axis only once in this interval.

This analysis was pivotal in the last step of the solution, establishing that the function meets at most one x-intercept, hence confirming the root’s uniqueness over the interval.