Problem 58 - Calculus Early Transcendentals

Question

From Exercise 57 , the polynomial \(f(x)=x^{3}+b x^{2}+1\) has one inflection point. Use a graphing utility to reach a conclusion about the effect of the constant \(b\) on the location of the inflection point. Use \(f^{\prime \prime}\) to explain what you have observed graphically.

Step-by-Step Solution

Answer

The inflection point is at \(x = -\frac{b}{3}\), moving linearly with \(b\).

Step 1: Find the Second Derivative

To find where the inflection points lie, we need the second derivative of the given polynomial function. Start with the function \(f(x) = x^3 + b x^2 + 1\). First, find the first derivative: \(f'(x) = 3x^2 + 2bx\). Now, find the second derivative: \(f''(x) = 6x + 2b\).

Step 2: Determine the Inflection Point Condition

A point of inflection occurs where the second derivative equals zero. Set \(f''(x) = 6x + 2b = 0\). Solve for \(x\) to find the inflection point: \[ 6x + 2b = 0 \] \[ 6x = -2b \] \[ x = -\frac{b}{3} \]

Step 3: Analyze the Effect of \(b\) on the Inflection Point

The solution \(x = -\frac{b}{3}\) indicates that the location of the inflection point varies linearly with \(b\). As \(b\) increases or decreases, the inflection point moves along the x-axis. When using a graphing utility and adjusting \(b\), you will observe that the inflection point shifts left for positive \(b\) and right for negative \(b\).

Step 4: Evaluate Graphical Observations

Graph the function \(f(x) = x^3 + bx^2 + 1\) for various values of \(b\) to visualize the shifting inflection point. Check the curvatures to see how the graph changes concavity at each calculated \(x\) value. The inflection point moves accordingly as \(b\) changes.

Key Concepts

Polynomial Function

Polynomial functions are key components in algebra and calculus that are formed from variables, coefficients, and exponents. A polynomial function is expressed in the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\). Here, each \(a_i\) is a constant coefficient, and \(n\) represents the degree of the polynomial, which is a whole number.

In the specific polynomial function from the exercise, \(f(x) = x^3 + bx^2 + 1\), \(x^3\) is the highest degree term, making it a cubic polynomial. The constant coefficient \(b\) influences the shape and position of the graph but doesn't affect the degree. This function tells us how variables are interacting with each other, and thus it forms a basis for analyzing more complex behaviors like those expressed in calculus.

Overall, polynomial functions provide mathematical models for many real-world situations, where trends and patterns need to be analyzed and interpreted.

Second Derivative

The second derivative of a function is a powerful tool used to determine the behavior of a graph, particularly in finding inflection points and understanding curvature. Start by taking the first derivative to find the rate of change, then differentiate again to get the second derivative.

For the given function \(f(x) = x^3 + bx^2 + 1\), the first derivative is \(f'(x) = 3x^2 + 2bx\). The second derivative, \(f''(x) = 6x + 2b\), helps us locate inflection points. An inflection point occurs where the second derivative changes sign, signaling a change in concavity.

Setting \(6x + 2b = 0\) and solving for \(x\) gives the inflection point condition: \(x = -\frac{b}{3}\). This equation illustrates how the constant \(b\) impacts the location, highlighting the importance of the second derivative in both theoretical and practical graph analysis.

Graphing Utility

A graphing utility is an indispensable tool in visualizing mathematical functions and analyzing their characteristics. It allows students and mathematicians to graph polynomial functions and observe changes dynamically.

By graphing \(f(x) = x^3 + bx^2 + 1\) for different values of \(b\), one can see how the inflection point's location varies along the x-axis. This visual approach complements the analytical solution obtained by solving the second derivative. Positive adjustments of \(b\) push the inflection point left, while negative ones push it right.

• Graphing utilities enhance understanding by providing visual evidence of mathematical concepts.
• They also allow exploration of a function's behavior in ways that purely algebraic methods might not.

Overall, using graphing utilities bridges the gap between computational algebra and intuitive understanding of function behavior.