Problem 250 - Calculus Volume 1

Question

Consider a third-degree polynomial \(f(x), \quad\) which has the properties \(f^{\prime}(1)=0, f^{\prime}(3)=0 .\) Determine whether the following statements are true or false. Justify your answer. If \(f(x)\) has one inflection point, then it has three real roots.

Step-by-Step Solution

Answer

False, a single inflection point doesn't guarantee three real roots.

Step 1: Identify Given Information

We are given a third-degree polynomial, \(f(x)\), and critical points \(f'(1) = 0\) and \(f'(3) = 0\). We are tasked with determining if the polynomial has three real roots if it has one inflection point.

Step 2: Understand Critical Points and Inflection Points

The critical points \(f'(x) = 0\) mean that the slopes of the tangent to \(f(x)\) are zero at those values of \(x\). An inflection point is where the concavity changes, which occurs when \(f''(x) = 0\). A third-degree polynomial can have at most one inflection point.

Step 3: Analyze the Third-Degree Polynomial

A third-degree polynomial \(f(x)\) is generally written as \(f(x) = ax^3 + bx^2 + cx + d\). Its first derivative is \(f'(x) = 3ax^2 + 2bx + c\), whose roots are the critical points.

Step 4: Use Given Critical Points in First Derivative

The critical points \(x = 1\) and \(x = 3\) are roots of the equation \(3ax^2 + 2bx + c = 0\). Since this is a quadratic, it means that the polynomial's turning points do not contribute to an inflection point directly, but rather identify local max or min.

Step 5: Determine the Number of Real Roots

Since there is only one inflection point possible in a third-degree function, it indicates that \(f''(x) = 0\) has a single real solution. The polynomial's real roots depend on how these stationary points affect the graph, which can be non-trivial, but crucially, it's impossible to conclude all roots based solely on one inflection point.

Key Concepts

Third-Degree Polynomial

A third-degree polynomial, also called a cubic polynomial, takes the mathematical form of \(f(x) = ax^3 + bx^2 + cx + d\). Here, the highest power of \(x\) in the equation is 3, indicating its degree. This type of polynomial is unique because:

• It can have up to three roots, which are values that satisfy the equation \(f(x) = 0\).
• It can have up to two critical points, where the derivative \(f'(x) = 0\).
• It may feature one inflection point, which is where the concavity of the graph changes.

Understanding the behavior of cubic polynomials is significant in equations since they can represent a wide range of functions with different shapes. These shapes are dictated by the signs and values of the coefficients \(a, b, c,\) and \(d\), which influence how the graph stretches, compresses, and shifts.

Critical Points

Critical points occur where the first derivative of a function equals zero. For the polynomial \(f(x) = ax^3 + bx^2 + cx + d\), the first derivative is \(f'(x) = 3ax^2 + 2bx + c\). By solving \(f'(x) = 0\), we identify these critical points, which indicate where the function's slope is zero.

• They are important in finding the local maxima and minima of a graph.
• In the given polynomial, the critical points are \(x = 1\) and \(x = 3\).

Finding critical points helps understand how the function behaves locally. This means pinpointing where the graph of the polynomial peaks or dips. For cubic polynomials, these points can signify turning points, which can influence the overall shape of the graph considerably.

Inflection Point

An inflection point is a unique feature of a graph where the concavity changes. For a cubic polynomial, this occurs where the second derivative equals zero. If you begin with \(f(x) = ax^3 + bx^2 + cx + d\), the second derivative is \(f''(x) = 6ax + 2b\).

• Solving \(f''(x) = 0\) will find the inflection points.
• Generally, a cubic polynomial can have only one inflection point.

This point is significant because it tells us about changes in the direction of the graph's curvature. Understanding inflection points can help with sketching the graph accurately and predicting how the function behaves between the critical points.

Real Roots

Real roots of a polynomial are the values of \(x\) that make the equation \(f(x) = 0\) equal zero. For a third-degree polynomial, there can be up to three real roots:

• Depending on the polynomial's coefficients, it may have one, two, or three real roots.
• These roots are the x-intercepts of the polynomial's graph.

It is crucial to understand:

• Just because a cubic polynomial has one inflection point does not guarantee it has three real roots.
• Additional information, such as critical points and specific coefficients, is needed to determine the number of real roots.

In the context of a cubic polynomial with certain critical points and possible inflection points, the graph's behavior between these points will affect the number of real roots. This means understanding not only the locations of these points but also the function's behavior around them.