Problem 60 - Calculus Early Transcendentals

Question

Find conditions on \(a, b, c,\) and \(d\) so that the graph of the polynomial \(f(x)=a x^{3}+b x^{2}+c x+d\) has (a) exactly two horizontal tangents (b) exactly one horizontal tangent (c) no horizontal tangents.

Step-by-Step Solution

Answer

(a) \(4b^2 - 12ac > 0\), (b) \(4b^2 - 12ac = 0\), (c) \(4b^2 - 12ac < 0\).

Step 1: Understanding Horizontal Tangents

Horizontal tangents occur where the derivative of the function is zero. For the given polynomial \( f(x) = ax^3 + bx^2 + cx + d \), we need to find its derivative \( f'(x) \) and set it equal to zero to find these points.

Step 2: Finding the Derivative

Compute the derivative of the function \( f(x) = ax^3 + bx^2 + cx + d \). We apply the power rule, obtaining \( f'(x) = 3ax^2 + 2bx + c \).

Step 3: Set Up the Condition for Horizontal Tangents

To find horizontal tangents, solve \( f'(x) = 3ax^2 + 2bx + c = 0 \). This is a quadratic equation. The number of solutions is determined by the discriminant \( \Delta = (2b)^2 - 4 \cdot 3a \cdot c \).

Step 4: (a) Two Horizontal Tangents

For two solutions to the equation, the discriminant must be positive. Thus, \( \Delta = 4b^2 - 12ac > 0 \). This is the condition for the polynomial to have exactly two horizontal tangents.

Step 5: (b) One Horizontal Tangent

To have exactly one horizontal tangent, the discriminant should be zero, indicating one repeated root. Thus, \( \Delta = 4b^2 - 12ac = 0 \).

Step 6: (c) No Horizontal Tangents

For no real solutions (no horizontal tangents), the discriminant must be negative. Thus, \( \Delta = 4b^2 - 12ac < 0 \).

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, encompassing operations of addition, subtraction, multiplication, and non-negative integer powers. For a polynomial such as \(f(x) = ax^3 + bx^2 + cx + d\), each term is a product of a coefficient and a power of \(x\). The highest power of \(x\) in the polynomial is called its degree. Polynomials are foundational in calculus and algebra because they are used to represent a wide array of functions.
Understanding polynomials is crucial as they form the basis for more complex mathematical models. They are generally expressed as:

• Constant term where the degree is zero (e.g., \(d\)).
• Linear term as \(cx\) where the degree is one.
• Quadratic term as \(bx^2\) where the degree is two.
• Cubic term as \(ax^3\) where the degree is three.

These terms collectively make a polynomial easy to manipulate and integrate or differentiate in calculus, making them versatile mathematical tools.

Derivatives

Derivatives are a fundamental concept in calculus representing the rate of change of a function with respect to a variable. For a polynomial, its derivative provides insight into the behavior of the function, such as identifying maxima, minima, and points of inflection. To find the derivative of our polynomial \(f(x) = ax^3 + bx^2 + cx + d\), we apply the power rule, yielding \(f'(x) = 3ax^2 + 2bx + c\).
This principle allows us to:

• Determine the slope of the tangent line at any point by substituting a particular \(x\) value into \(f'(x)\).
• Find critical points by setting \(f'(x) = 0\) and solving the equation.

Knowing how to derive ensures students can thoroughly analyze how a function behaves in different segments of its domain and solve practical problems involving rates and trends.

Tangent Lines

Tangent lines are straight lines that touch a curve at exactly one point, representing the instantaneous slope or rate of change of the curve at that point. The slope of the tangent line to a function \(f(x)\) at a specific point is given by its derivative \(f'(x)\).
Horizontal tangents specifically affect the analysis of polynomial functions, as seen in our example, where identifying horizontal tangents involves solving \(f'(x) = 0\).
In calculus, analyzing tangent lines helps in understanding the following:

• Maxima and minima, where horizontal tangents occur.
• The direction and rate of change of the curve at various points.
• Identifying equilibrium points in various mathematical and physical systems.

This concept is vital, as it allows for predictions about function behavior and provides a basis for optimizing performance in engineering and physics applications.

Quadratic Equation

The quadratic equation forms an integral part of solving for horizontal tangents in polynomials. By setting the derivative \(f'(x) = 3ax^2 + 2bx + c\) to zero, a quadratic equation emerges: \(3ax^2 + 2bx + c = 0\). This form delivers solutions that determine where the polynomial might have horizontal tangent lines.
Quadratic equations are significant in the following ways:

• They allow solvers to determine if a polynomial has horizontal tangents by analyzing the discriminant \(\Delta = 4b^2 - 12ac\).
• They categorize the nature of roots, where a positive discriminant implies two solutions (two horizontal tangents), zero indicates a double root (one horizontal tangent), and negative indicates no real solutions (no horizontal tangents).

Mastery of quadratic equations enables students to interpret and solve for the characteristics of a polynomial graph, leveraging this knowledge in various scientific and engineering fields.