Problem 50 - Calculus A New Horizon

Question

From Exercise \(49,\) 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 moves horizontally as \(b\) changes, located at \(x = -\frac{b}{3}\).

Step 1: Identify the Inflection Point

An inflection point occurs where the second derivative of the function changes sign. For the polynomial \(f(x) = x^3 + b x^2 + 1\), we first need to determine its second derivative \(f''(x)\).

Step 2: Compute the First Derivative

The first derivative of \(f(x) = x^3 + b x^2 + 1\) is found using the power rule: \(f'(x) = 3x^2 + 2bx\).

Step 3: Compute the Second Derivative

Take the derivative of \(f'(x) = 3x^2 + 2bx\) to get the second derivative. So, \(f''(x) = 6x + 2b\).

Step 4: Find the Inflection Point

Set the second derivative to zero to find the potential inflection points: \(6x + 2b = 0\). Solve for \(x\) to find \(x = -\frac{b}{3}\). This is where \(f''(x)\) changes sign.

Step 5: Analyze Graphically with b Variation

Using a graphing utility, plot the function \(f(x)\) for various values of \(b\). Observe how the inflection point \(x = -\frac{b}{3}\) shifts horizontally based on the value of \(b\).

Step 6: Conclusion from Graphical Observation

The graph should confirm that as \(b\) increases, the inflection point moves to the left, and as \(b\) decreases, it moves to the right. The movement of the inflection point is consistent with the equation \(x = -\frac{b}{3}\).

Key Concepts

Second Derivative

The second derivative is a fundamental concept when analyzing the shape and curvature of functions. For any given function, the second derivative provides information about the concavity of the graph.
For the polynomial function given by \[ f(x) = x^3 + b x^2 + 1 \] its second derivative is calculated by differentiating the first derivative:

• First derivative: \[ f'(x) = 3x^2 + 2bx \]
• Second derivative: \[ f''(x) = 6x + 2b \]

The second derivative, \( f''(x) = 6x + 2b \), indicates how the slope of the tangent line changes as \( x \) changes. The inflection point, where \( f''(x) = 0 \), is crucial as it's the point where the concavity switches, meaning it goes from concave up to concave down, or vice versa. Solving \( 6x + 2b = 0 \) gives us the inflection point \( x = -\frac{b}{3} \), where the second derivative changes its sign.

Graphing Utility

A graphing utility is an excellent tool for visualizing the behavior of functions and their derivatives. By utilizing a graphing calculator or software, we can quickly see how different components of a polynomial function affect its graph.
When analyzing the polynomial \( f(x) = x^3 + b x^2 + 1 \) and its inflection point, the graphing utility provides a visual representation of how the changes in the constant \( b \) impact the function. By plotting various values of \( b \), we can observe:

• The horizontal shift of the inflection point as \( b \) changes
• The consistent movement pattern, following \( x = -\frac{b}{3} \)

This visual method not only validates our analytical findings but also strengthens our understanding by connecting algebraic results with graphical behaviors.

Polynomial Function

Polynomial functions are mathematical expressions consisting of variables and coefficients, structured in terms of powers of the variables. They are fundamental to many areas of mathematics and serve as building blocks for more complex functions.
The specific polynomial in this exercise, \( f(x) = x^3 + b x^2 + 1 \), is a cubic polynomial. Here:

• The term \( x^3 \) makes it cubic, giving it a maximum of two turning points and one possible inflection point.
• The coefficient \( b \) controls the position and shape of the graph, particularly affecting the inflection point.
• The constant term \( 1 \) shifts the graph up or down vertically, without changing its basic shape.

Understanding polynomials is crucial for deriving their properties, such as roots, maxima, minima, and inflection points, and determining how additional parameters like \( b \) in this case influence their graphical representation.

Derivative Analysis

Derivative analysis provides insights into the behavior of functions, capturing their rate of change and helping to identify critical points such as maxima, minima, and inflection points. This process begins with determining the first derivative and then the second derivative.
For the function \( f(x) = x^3 + b x^2 + 1 \), derivative analysis follows these steps:

• First, compute \( f'(x) = 3x^2 + 2bx \), which describes the slope or tangent at any point \( x \).
• Second, compute \( f''(x) = 6x + 2b \). It provides information about the acceleration of the slope—how quickly it is increasing or decreasing.
• Identify the inflection point by setting \( f''(x) = 0 \) and solving for \( x \), leading to \( x = -\frac{b}{3} \).

Through derivative analysis, we gain a deeper understanding of the function's geometric properties. We find specific locations on the graph where the function alters its concavity, which is essential for both theoretical insights and practical applications.