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: Express the Cubic Function

Given the cubic function, write it in terms of its roots: \[ f(x) = a(x - r_1)(x - r_2)(x - r_3) \] where \( a \) is a constant and \( r_1, r_2, r_3 \) are the roots of the function.

Step 2: Find the Derivative of the Function

Find the first derivative of \( f(x) \) using the product rule:\[ f'(x) = a \left[ (x - r_2)(x - r_3) + (x - r_1)(x - r_3) + (x - r_1)(x - r_2) \] with respect to \( x \). This simplifies the derivative to:\[ f'(x) = a(3x^2 - 2x(r_1 + r_2 + r_3) + (r_1 r_2 + r_2 r_3 + r_1 r_3)) \]

Step 3: Find the Second Derivative

Differentiate \( f'(x) \) to find \( f''(x) \), which is needed to find the inflection point:\[ f''(x) = a(6x - 2(r_1 + r_2 + r_3)) \]

Step 4: Set the Second Derivative to Zero

Set \( f''(x) = 0 \) to determine the \( x \)-coordinate of the inflection point:\[ 6x - 2(r_1 + r_2 + r_3) = 0 \] Solve for \( x \):\[ 6x = 2(r_1 + r_2 + r_3) \]\[ x = \frac{r_1 + r_2 + r_3}{3} \]

Step 5: Conclude the Result

We have shown that the \( x \)-coordinate of the inflection point is indeed \( \frac{r_1 + r_2 + r_3}{3} \), as required.

Key Concepts

Inflection Point

Understanding the concept of an inflection point is crucial when analyzing the behavior of a cubic function. An inflection point is where the curve changes its concavity; that is, it moves from being concave up to concave down, or vice versa. This is a point on the curve where the curvature changes sign. To identify an inflection point in mathematical terms, we look for where the second derivative of the function equals zero, indicating a possible change in concavity. By examining the cubic function of form \[ f(x) = a(x - r_1)(x - r_2)(x - r_3) \], we found that setting the second derivative to zero helps us find the x-coordinate \[ x = \frac{r_1 + r_2 + r_3}{3} \], where the inflection point occurs.

Real Zeros

Real zeros of a function are the x-values that make the function equal to zero. They are also called roots or solutions of the equation. For a cubic function like \[ f(x) = a(x - r_1)(x - r_2)(x - r_3) \], the real zeros are

• \( r_1 \)
• \( r_2 \)
• \( r_3 \)

These zeros are the points at which the graph of the function intersects the x-axis. In the context of finding the inflection point, knowing these real zeros helps in expressing the function and its derivatives more efficiently. The symmetry and distribution of these roots are essential for understanding the overall shape and nature of the cubic function.

Second Derivative

The second derivative of a function provides information about the concavity of the function. To find the second derivative of a cubic function, we first need to determine the first derivative and then differentiate it again. For example, if \[ f'(x) = a(3x^2 - 2x(r_1 + r_2 + r_3) + (r_1 r_2 + r_2 r_3 + r_1 r_3)) \], then the second derivative is \[ f''(x) = a(6x - 2(r_1 + r_2 + r_3)) \]. Setting \[ f''(x) \] to zero will yield the x-value of the inflection point. This step is essential as it helps in revealing not only the inflection points but also contributes to understanding the rate of change of the function's slope.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. If you have a function written as a product of two or more smaller functions, you can differentiate it using this rule. For a cubic function \[ f(x) = a(x - r_1)(x - r_2)(x - r_3) \], the product rule is essential. It allows us to find the first derivative smoothly. The rule states that \[ (uv)' = u'v + uv' \], where \( u \) and \( v \) are functions of \( x \). By applying the product rule to the quadratic factors in the cubic expression, we derive the first derivative successfully. This step is instrumental in progressing towards finding the second derivative and discovering the inflection point.