Problem 74 - Thomas Calculus in SI Units

Question

Consider the cubic function $$f(x)=a x^{3}+b x^{2}+c x+d$$ a. Show that \(f\) can have \(0,1,\) or 2 critical points. Give examples and graphs to support your argument. b. How many local extreme values can \(f\) have?

Step-by-Step Solution

Answer

A cubic function can have 0, 1, or 2 critical points, resulting in 0, 1, or 2 local extrema.

Step 1: Identifying critical points

To find the critical points of the function \(f(x) = ax^3 + bx^2 + cx + d\), we take its derivative: \(f'(x) = 3ax^2 + 2bx + c\). Critical points occur where the derivative is zero, so we need to solve the equation \(3ax^2 + 2bx + c = 0\). This is a quadratic equation.

Step 2: Analyzing the quadratic solution

The number of critical points depends on the number of real solutions to the quadratic equation \(3ax^2 + 2bx + c = 0\). The discriminant of this quadratic is given by \(\Delta = (2b)^2 - 4(3a)(c) = 4b^2 - 12ac\).

Step 3: Determining the number of real solutions

1. If \(\Delta > 0\), the quadratic has two distinct real solutions, leading to two critical points. 2. If \(\Delta = 0\), the quadratic has one real solution (a repeated root), leading to one critical point. 3. If \(\Delta < 0\), there are no real solutions, so no critical points exist.

Step 4: Examples of cubic functions

For \(f(x) = x^3 - 3x + 2\), the discriminant of \(f'(x) = 3x^2 - 3\) is positive, giving two critical points. For \(f(x) = x^3\), the discriminant of \(f'(x) = 3x^2\) is zero, resulting in one critical point at \(x = 0\). For \(f(x) = x^3 + x\), \(f'(x) = 3x^2 + 1\) has no real roots, showing no critical points.

Step 5: Local extreme values analysis

A cubic function can have a maximum of one local maximum and one local minimum, which correspond to the two distinct critical points when \(\Delta > 0\). If there is only one critical point, it might be a point of inflection, meaning no local extrema occur. Thus, \(f(x)\) can have 0, 1, or 2 local extrema based on its critical points.

Key Concepts

Critical Points in Cubic Functions

A critical point of a function is a point on the graph where the derivative is zero or undefined, and it's crucial for determining the function's behavior. For our cubic function, given by \( f(x) = ax^3 + bx^2 + cx + d \), the critical points are found by taking the derivative: \( f'(x) = 3ax^2 + 2bx + c \). A critical point exists where this derivative equals zero, meaning it depends on the number of solutions to the equation \( 3ax^2 + 2bx + c = 0 \). Some key points to remember about critical points in cubic functions:

• The function can have up to two critical points due to its quadratic nature. This is because a quadratic equation can have zero, one, or two real solutions.
• The discriminant \( \Delta = 4b^2 - 12ac \) determines the number of these solutions:
• If \( \Delta > 0 \), there are two distinct solutions, resulting in two critical points.
• If \( \Delta = 0 \), there is one repeated solution, indicating one critical point.
• If \( \Delta < 0 \), there are no real solutions, meaning there are no critical points.

Understanding these points is fundamental to analyzing the graph's structure, helping to pinpoint where the function could possibly change direction.

Local Extrema

Local extrema are the points where a function achieves local maximum or minimum values. In other words, these are the peaks or valleys of the graph. For cubic functions like \( f(x) = ax^3 + bx^2 + cx + d \), local extrema occur at critical points where the derivative changes sign. Here's what to remember about local extrema of cubic functions:

• A cubic function can have up to two local extrema, depending on the number of critical points.
• When there are two critical points (\( \Delta > 0 \)), these can be a local maximum and a local minimum.
• If there is only one critical point (\( \Delta = 0 \)), it might be a point of inflection—a single point where the curve changes concavity.
• If there are no critical points (\( \Delta < 0 \)), the cubic function does not have any local extrema.

In summary, the presence and nature of critical points directly influence whether a cubic function has local extrema and how many. Studying these helps determine the nature of the cubic graph and any significant features it presents.

Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \). In the context of cubic functions, such as \( f(x) = ax^3 + bx^2 + cx + d \), we deal with quadratic equations when finding critical points. The derivative \( f'(x) = 3ax^2 + 2bx + c \) is quadratic, and solutions to \( 3ax^2 + 2bx + c = 0 \) reveal the critical points of the cubic function. Key characteristics of the quadratic equation include:

• Roots: These are the solutions to the equation. The number and type of roots depend on the discriminant \( \Delta = b^2 - 4ac \).
• Discriminant Analysis:
  • If \( \Delta > 0 \), the equation has two distinct real roots.
  • If \( \Delta = 0 \), there is one real root (a repeated root).
  • If \( \Delta < 0 \), there are no real roots—but two complex roots instead.
• Critical Points: The roots of the quadratic equation, when real, are the critical points of the cubic function.

Understanding how to solve a quadratic equation and analyze its discriminant is an essential skill for investigating the nature of critical points in cubic functions. This knowledge aids in predicting the behavior and structure of these functions effectively.