Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that the cubic function \(f(x)\) has three real zeros, \(r_{1}, r_{2}\), and \(r_{3}\). Show that its inflection point has \(x\) -coordinate \(\left(r_{1}+r_{2}+r_{3}\right) / 3 .\) Hint: \(f(x)=a\left(x-r_{1}\right)\left(x-r_{2}\right)\left(x-r_{3}\right)\)

Step-by-Step Solution

Answer

The inflection point has an \(x\)-coordinate of \(\frac{r_1 + r_2 + r_3}{3}\).

Step 1: Expand the Cubic Function

Start with the given function form: \(f(x) = a(x - r_1)(x - r_2)(x - r_3)\). Expand this expression to get it in the standard form of a cubic polynomial. First, expand two of the factors: \[(x - r_1)(x - r_2) = x^2 - (r_1 + r_2)x + r_1r_2\] Then, multiply this result by the third factor \((x - r_3)\): \[a(x^2 - (r_1 + r_2)x + r_1r_2)(x - r_3)\]. Continued expansion gives: \[ax^3 - a(r_1 + r_2 + r_3)x^2 + a(r_1r_2 + r_2r_3 + r_3r_1)x - ar_1r_2r_3\]. Thus, the expanded form is: \[f(x) = ax^3 - a(r_1 + r_2 + r_3)x^2 + a(r_1r_2 + r_2r_3 + r_3r_1)x - ar_1r_2r_3\].

Step 2: Find the Derivative and Second Derivative

To find the inflection point, compute the second derivative of \(f(x)\). First, find the first derivative: \(f'(x) = 3ax^2 - 2a(r_1 + r_2 + r_3)x + a(r_1r_2 + r_2r_3 + r_3r_1)\). Then, the second derivative is \(f''(x) = 6ax - 2a(r_1 + r_2 + r_3)\).

Step 3: Set the Second Derivative to Zero

The inflection point occurs where the second derivative is zero. Set \(f''(x) = 0\): \(6ax - 2a(r_1 + r_2 + r_3) = 0\). Simplify this equation: \[6ax = 2a(r_1 + r_2 + r_3)\].

Step 4: Solve for x

Divide both sides of \(6ax = 2a(r_1 + r_2 + r_3)\) by \(6a\) to solve for \(x\): \[x = \frac{2a(r_1 + r_2 + r_3)}{6a} = \frac{r_1 + r_2 + r_3}{3}\]. This is the \(x\)-coordinate of the inflection point.

Key Concepts

Cubic Polynomial

A cubic polynomial is an expression of the form \(ax^3 + bx^2 + cx + d\), where \(a, b, c, \) and \(d\) are constants, and \(a eq 0\). It is called "cubic" because of the highest power of the variable, which is 3. This highest degree term \(ax^3\) is what gives the polynomial its unique characteristics and behavior. Cubic polynomials have up to three real roots or "zeros," which are the values of \(x\) that make the polynomial equal to zero. These roots can be found by factoring the polynomial, applying the Rational Root Theorem, or using numerical methods for complicated cases. A distinctive feature of cubic functions is their ability to change curvature, leading to points of inflection.

Zeros of Polynomial

The zeros of a polynomial are the solutions to the equation when the polynomial is set equal to zero. In simpler terms, they are the \(x\)-values where the graph of the polynomial intersects the \(x\)-axis. For a cubic polynomial, there can be up to three real zeros. These zeros not only help in graphing the polynomial but also play a significant role in its factorization.For example, if a cubic polynomial is expressed as \(f(x) = a(x - r_1)(x - r_2)(x - r_3)\), then \(r_1, r_2, \) and \(r_3\) are the zeros of the polynomial. These values are critical in understanding the polynomial's behavior and can be used to find other features of the graph, such as the inflection point.

Inflection Point

An inflection point is where the curvature of the polynomial graph changes direction. For a cubic function \(f(x)\), this is where the second derivative \(f''(x)\) is equal to zero. The second derivative helps determine the concavity of a function's graph. At the inflection point, the graph transitions from being concave up (curving upwards) to concave down (curving downwards), or vice versa.In the case of cubic polynomials, the inflection point has an interesting property: its \(x\)-coordinate is the average of its roots. Given the roots \(r_1, r_2, \) and \(r_3\), the inflection point \(x\)-coordinate can be calculated by the formula \(x = \frac{r_1 + r_2 + r_3}{3}\). This position on the \(x\)-axis is where the graph changes its curvature.

Derivative Calculation

Calculating the derivatives of a cubic polynomial is essential to determine its critical points and inflection points. The first derivative \(f'(x)\) of a cubic polynomial \(f(x) = ax^3 + bx^2 + cx + d\) is calculated as \(f'(x) = 3ax^2 + 2bx + c\). This derivative helps to find points where the slope of the tangent line is zero, which are potential local maxima or minima.The second derivative, \(f''(x) = 6ax + 2b\), is used to identify the inflection point. Setting \(f''(x) = 0\) and solving for \(x\) provides the \(x\)-coordinate of the inflection point. This is where the graph changes concavity. Understanding these calculations allows deeper insights into the behavior of the polynomial and its graph.