Problem 19 - Essential Calculus Early Transcendentals

Question

Show that the equation \(x^{3}-15 x+c=0\) has at most one root in the interval \([-2,2] .\)

Step-by-Step Solution

Answer

The function has at most one root in the interval \([-2, 2]\) because it is strictly decreasing there.

Step 1: Identify the Function and Interval

We are given the function \(f(x) = x^3 - 15x + c\) and need to show it has at most one root in the interval \([-2, 2]\). A root occurs where \(f(x) = 0\).

Step 2: Find the Derivative

Calculate the derivative of \(f(x)\), which helps us determine where the function is increasing or decreasing. The derivative is \(f'(x) = 3x^2 - 15\).

Step 3: Analyze the Derivative

Set \(f'(x) = 0\) to find critical points: \(3x^2 - 15 = 0\). Solving for \(x\), we get \(x^2 = 5\) or \(x = \pm \sqrt{5}\). The values \(\pm \sqrt{5}\) are approximately \(\pm 2.236\), which are outside the interval \([-2, 2]\).

Step 4: Determine Behavior in the Interval

Since there are no critical points within \([-2, 2]\), \(f'(x) > 0\) for all \(x\) in the interval because \(3(-2)^2 - 15 = 12 - 15 = -3\), and \(3(2)^2 - 15 = 12 - 15 = -3\), both of which are negative. Therefore, \(f(x)\) is decreasing in \([-2, 2]\).

Step 5: Conclude on Number of Roots

Since \(f(x)\) is strictly decreasing in \([-2, 2]\) and there are no critical points in this interval, \(f(x)\) can cross the x-axis at most once, implying at most one root exists in this interval.

Key Concepts

Critical Points

Critical points of a function are values of the independent variable, typically noted as \(x\), where the first derivative, \(f'(x)\), is zero or undefined. These points can indicate where a function's graph has peaks, valleys, or inflection points.

In this particular problem, we start by calculating the derivative of the function \(f(x) = x^3 - 15x + c\) to find these points. The derivative is \(f'(x) = 3x^2 - 15\).

Next, we set the derivative equal to zero: \(3x^2 - 15 = 0\). Solving for \(x\), we find \(x^2 = 5\). Thus, \(x = \pm \sqrt{5}\), which calculates to approximately \(\pm 2.236\).

Since \(\pm 2.236\) are not within the interval \([-2, 2]\), the function has no critical points within the specified interval. This means the function doesn't switch from increasing to decreasing or vice versa within this range.

Interval Analysis

Interval analysis involves examining the behavior of a function over a specific range of \(x\)-values. It helps us understand if the function is increasing, decreasing, and how many roots it may possess.

For the given function \(f(x) = x^3 - 15x + c\), we consider the interval \([-2, 2]\). We are interested in verifying how the function behaves over this range and determining the presence of any roots.

Since the critical points \(x = \pm \sqrt{5}\) are outside the interval, \(f(x)\) has no changes in direction in the interval \([-2, 2]\). This information is crucial, as it indicates that \(f(x)\) is either entirely increasing or decreasing across this interval.

This step of analysis supports the conclusion that \(f(x)\) cannot have more than one root in the given range, as a change in direction would be needed to possess more than one root.

Derivative Analysis

Derivative analysis is key in understanding the nature of a function's increase or decrease over an interval. By examining \(f'(x)\), we gain insights into how a function's graph behaves.

For our function, \(f'(x) = 3x^2 - 15\). We found that the critical points don't fall within the interval \([-2, 2]\). However, evaluating \(f'(x)\) at a point in the interval, like \(-2\) or \(2\), gives \(f'(-2) = 12 - 15 = -3\) and \(f'(2) = 12 - 15 = -3\). This negative value indicates that the function is decreasing over the entire interval.

Since \(f(x)\) is consistently decreasing, it suggests that the function can cross the x-axis at most once within the interval, confirming there can be at most one root. This analysis is crucial for students to gain proficiency in predicting the number and nature of roots.