Problem 106 - Algebra and Trigonometry

Question

Exercises 105–107 will help you prepare for the material covered in the next section. If \(f(x)=x^{3}-2 x-5,\) find \(f(2)\) and \(f(3) .\) Then explain why the continuous graph of \(f\) must cross the \(x\) -axis between 2 and 3

Step-by-Step Solution

Answer

By substituting the values \`x=2\` and \`x=3\` into the function \( f(x)=x^{3}-2 x-5\), we will find that \(f(2)=-1\) and \(f(3)=16\). Given that \(f(2)<0\) and \(f(3)>0\), and the function is continuous, we can apply the Intermediate Value Theorem which leads to the conclusion that the graph of \( f \) must cross the x-axis between 2 and 3.

Step 1: Calculating the Value of the Function at x=2

We start by substituting x=2 in the function \(f(x)=x^{3}-2 x-5\). This will give us \(f(2)=(2)^{3}-2*(2)-5=8-4-5=-1\)

Step 2: Calculating the Value of the Function at x=3

We then substitute x=3 in the function \(f(x)=x^{3}-2 x-5\). This will give us \(f(3)=(3)^{3}-2*(3)-5=27-6-5=16\)

Step 3: Applying the Intermediate Value Theorem

Since \(f(2)=-1<0\) and \(f(3)=16>0\), there exists a point c in [2, 3] such that \(f(c)=0\). Which means that the graph of \(f\) must cross the x-axis between 2 and 3 as the function \(f\) is continuous and is changing sign between these points

Key Concepts

Polynomial Functions

Polynomial functions are foundational mathematical expressions composed of variables and coefficients, arranged in the form of powers. For example, consider the polynomial function given in the exercise: \[ f(x) = x^{3} - 2x - 5 \] This is a cubic polynomial, characterized by the highest exponent of 3. Polynomial functions can vary widely, from linear (e.g., \(f(x) = x + 1\)) to quadratic (e.g., \(f(x) = x^{2} - x + 4\)), and higher degrees such as cubic, quartic, and beyond. Important features of polynomial functions include:

• They are algebraic expressions where the degree denotes the highest exponent.
• They can have one or more roots, which are the values of \(x\) where the function equals zero.
• They exhibit a smooth, continuous curve when graphed.
• They can be positive, negative, or zero at different values of \(x\).

In our exercise, finding the values of \(f(2)\) and \(f(3)\) helps in assessing the behavior of the polynomial and sets the stage for applying the Intermediate Value Theorem.

Continuous Functions

Continuous functions are those that have no breaks, jumps, or holes when plotted on a graph. They ensure that every value in a domain has a corresponding point on the graph, making them predictable and smooth. In the context of solving our exercise, the function \(f(x) = x^{3} - 2x - 5\) is continuous, which means as \(x\) changes from 2 to 3, \(f(x)\) transitions without interruption. Continuous functions possess the following properties:

• They can be drawn without lifting the pen from the paper.
• For any given interval \([a, b]\), if a function is continuous, any \(f(c)\) must exist within the interval.
• They maintain defined limits at every point within their domain.

In our exercise, the core understanding is that the polynomial, being continuous, allows us to utilize the Intermediate Value Theorem. This theorem states that if \(f(x)\) changes sign over an interval, it must cross the x-axis at some point within that interval.

Root Finding

Root finding is the process of determining where a function crosses the x-axis, i.e., where \(f(x) = 0\). For polynomial functions, this involves identifying the values of \(x\) that satisfy this condition. In our exercise, the task is to find the root of the function \(f(x) = x^{3} - 2x - 5\) between the interval \[2, 3\]. The steps to find a root include:

• Evaluate the function at specific points within an interval. For the problem, \(f(2)\) and \(f(3)\) were calculated to be \(-1\) and \(16\), respectively.
• Apply the Intermediate Value Theorem, which guarantees a root exists if a function is continuous over an interval and changes sign. Here, since \(f(2) < 0\) and \(f(3) > 0\), \(f(x)\) must equal zero at some point between 2 and 3.
• Use numerical methods or graphical analysis to approximate the root within the given interval.

Root finding is essential in calculus and algebra as it aids in solving equations of varying complexity, helping us understand the function's behavior around that point.