Problem 60 - Calculus - AP Edition

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: Define Horizontal Tangent Condition

For a horizontal tangent, the derivative of the function must be equal to zero. Compute the derivative of the polynomial: \( f'(x) = 3ax^2 + 2bx + c \). The condition for a horizontal tangent at a point is \( f'(x) = 0 \).

Step 2: Find Condition for Two Horizontal Tangents

Two horizontal tangents indicate that the derivative \( f'(x) = 3ax^2 + 2bx + c \) must have two distinct real roots. This will happen if the discriminant of this quadratic equation is positive. The discriminant is \( \Delta = (2b)^2 - 4(3a)(c) = 4b^2 - 12ac \). For exactly two horizontal tangents, \( 4b^2 - 12ac > 0 \).

Step 3: Determine Condition for One Horizontal Tangent

To have exactly one horizontal tangent, the quadratic \( 3ax^2 + 2bx + c \) must have exactly one real root. This occurs when the discriminant is zero. Hence, \( 4b^2 - 12ac = 0 \) is the condition for exactly one horizontal tangent.

Step 4: Establish Condition for No Horizontal Tangents

No horizontal tangents imply that \( f'(x) = 3ax^2 + 2bx + c \) has no real roots, meaning the discriminant is negative. Thus, for no horizontal tangents, we need \( 4b^2 - 12ac < 0 \).

Key Concepts

Horizontal Tangents

A horizontal tangent on the graph of a function occurs where the slope of the tangent line is exactly zero. To find such points, we first look at the derivative of the function, since the derivative gives us the slope of the tangent line at any point along the curve. For our polynomial function, the derivative is given by:

• \( f'(x) = 3ax^2 + 2bx + c \)

When this expression is equal to zero, we find the points where the horizontal tangents occur. These points are critical for analyzing the nature of the function's graph, such as identifying peaks, valleys, or flat sections.

If this derivative equation has solutions for \(x\), then at those values, the graph of the original function \(f(x)\) will have horizontal tangents. The nature (number and multiplicity) of these solutions helps us determine various possible behaviors of the polynomial's graph.

Quadratic Discriminant

To better understand the roots of a quadratic equation such as the derivative, the quadratic discriminant is used. The discriminant of a quadratic equation of the form \(ax^2 + bx + c = 0\) is given by:

• \( \Delta = b^2 - 4ac \)

The discriminant helps in determining the nature and number of roots of the quadratic. Here's how it works:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root, meaning the graph of the quadratic just touches the x-axis at one point – often referred to as a "double root."
• If \( \Delta < 0 \), there are no real roots.

In the context of polynomial derivatives, applying the discriminant to \( 3ax^2 + 2bx + c \) helps us determine the number of horizontal tangents on the graph. We substitute the appropriate coefficients into the discriminant formula to understand when and how these tangents occur.

Real Roots

Identifying real roots involves finding the values of \(x\) that solve the derivative equation \( f'(x) = 0 \). These solutions (roots) correspond to points where horizontal tangents are possible on the graph of the polynomial function.

• When the discriminant is positive \( (4b^2 - 12ac > 0) \), the derivative has two distinct real roots, indicating two separate points where horizontal tangents occur.
• When the discriminant is zero \( (4b^2 - 12ac = 0) \), there is exactly one real root, signifying a single horizontal tangent point. This often correlates with a graph that "bounces off" the x-axis at the tangential contact.
• With a negative discriminant \( (4b^2 - 12ac < 0) \), no real roots exist, meaning the derivative never equals zero, and thus, no horizontal tangents exist on the graph.

Understanding these roots provides insight into the critical characteristics of the curve, helping one predict its general shape and behavior across the x-axis. It assists in visualizing how the curve might rise and fall, and where it flattens out if only briefly.