Problem 59 - Essential Calculus Early Transcendentals

Question

Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three \(x\) -intercepts \(x_{1}, x_{2},\) and \(x_{3},\) show that the \(x\) -coordinate of the inflection point is \(\left(x_{1}+x_{2}+x_{3}\right) / 3.\)

Step-by-Step Solution

Answer

A cubic function always has exactly one inflection point at \((x_1 + x_2 + x_3)/3\).

Step 1: Define the Cubic Function

Consider the general form of a cubic function which is usually written as \( f(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants.

Step 2: Find the Second Derivative

The point of inflection occurs where the second derivative changes sign. First, find the first derivative: \( f'(x) = 3ax^2 + 2bx + c \). Then, find the second derivative: \( f''(x) = 6ax + 2b \).

Step 3: Set the Second Derivative to Zero

To find the inflection point, set \( f''(x) = 0 \), which gives \( 6ax + 2b = 0 \). Solve for \( x \) to find the inflection point location: \( x = -\frac{b}{3a} \).

Step 4: Relate to the x-Intercepts

Assume \( f(x) = a(x-x_1)(x-x_2)(x-x_3) \) using the roots as intercepts. Expand to match the standard form: \( ax^3 - a(x_1 + x_2 + x_3)x^2 + a(x_1x_2 + x_2x_3 + x_3x_1)x - ax_1x_2x_3 \).

Step 5: Compare and Extract Coefficients

Compare the expanded form with \( ax^3 + bx^2 + cx + d \). The coefficient of \( x^2 \) in both expressions corresponds, so \( b = -a(x_1 + x_2 + x_3) \).

Step 6: Calculate Inflection Point Coordinate

Substitute \( b = -a(x_1 + x_2 + x_3) \) into the expression \( x = -\frac{b}{3a} \) from Step 3: \[ x = -\frac{-a(x_1 + x_2 + x_3)}{3a} = \frac{x_1 + x_2 + x_3}{3} \]. Thus, the point of inflection occurs at \( x = \frac{x_1 + x_2 + x_3}{3} \).

Key Concepts

Inflection Point in a Cubic Function

An inflection point is where a curve changes its concavity. In simpler terms, it's the spot where the curve shifts from bending upwards to downwards or vice versa. For a cubic function, represented by a third-degree polynomial, an inflection point is always present. This point can be identified by locating where the second derivative of the function changes sign.

For cubic functions like \( f(x) = ax^3 + bx^2 + cx + d \), the inflection point occurs when the second derivative, \( f''(x) \), equals zero. Through calculations, you will find that for a cubic function, there is only one such point where this transition happens, as given by \( x = -\frac{b}{3a} \). This mathematical property of cubic functions makes it straightforward to predict and locate the inflection point.

Understanding the Second Derivative

The second derivative of a function, symbolized as \( f''(x) \), provides insights into the curvature of the graph. For cubic functions, which typically take the form \( f(x) = ax^3 + bx^2 + cx + d \), the second derivative will tell you where the graph changes from concave up (like a cup) to concave down (like a cap), or vice versa.

When you find the first derivative from the cubic function, you'll get \( f'(x) = 3ax^2 + 2bx + c \). By differentiating it again, you obtain \( f''(x) = 6ax + 2b \). To find points of inflection, you solve \( 6ax + 2b = 0 \). This approach reveals the \( x \)-coordinate of the inflection point, showing where changes in curvature occur, thus making it a critical insight for understanding the overall behavior of the polynomial function's graph.

x-Intercepts and Their Role in Inflection Points

Cubic functions often have the beauty of being able to cross the \( x \)-axis up to three times, providing what are called \( x \)-intercepts. These intercepts are essentially the roots of the function, meaning the values of \( x \) that make the function zero.

Let's say a cubic function is given as \( f(x) = a(x-x_1)(x-x_2)(x-x_3) \). Here, \( x_1, x_2, x_3 \) are the \( x \)-intercepts. These intercepts provide essential information that can be used to locate the inflection point. By expanding this function, you can correlate it with the general form \( ax^3 + bx^2 + cx + d \), allowing you to derive that \( b = -a(x_1 + x_2 + x_3) \).

This relationship is crucial because it simplifies the location of the cubic function's inflection point. As indicated in the solution, the \( x \)-coordinate of the inflection point is \( \frac{x_1 + x_2 + x_3}{3} \). This shows a profound connection between the roots or intercepts of the cubic function and where the point of inflection sits.