Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the curve \(y=a x^{3}+b x^{2}+c x+d\) where \(a \neq 0\), has exactly one inflection point.

Step-by-Step Solution

Answer

The curve has exactly one inflection point at \(x = -\frac{b}{3a}\).

Step 1: Understand the Inflection Point

An inflection point is a point on a curve where the curvature changes sign. This is equivalent to the second derivative of the function changing sign at that point. Therefore, our task is to find the second derivative and solve for when it equals zero, indicating a potential inflection point.

Step 2: Compute the First Derivative

Start by finding the first derivative of the curve with respect to \(x\). The function is given as \(y=ax^3 + bx^2 + cx + d\). Differentiating this with respect to \(x\) gives: \[y' = \frac{d}{dx}(ax^3 + bx^2 + cx + d) = 3ax^2 + 2bx + c.\]

Step 3: Compute the Second Derivative

Differentiate the first derivative \(y' = 3ax^2 + 2bx + c\) with respect to \(x\) to find the second derivative:\[y'' = \frac{d}{dx}(3ax^2 + 2bx + c) = 6ax + 2b.\]

Step 4: Solve for the Inflection Point

Set the second derivative equal to zero to find potential inflection points:\[6ax + 2b = 0.\]Solve for \(x\):\[x = -\frac{b}{3a}.\]

Step 5: Verify the Single Inflection Point

The solution \(x = -\frac{b}{3a}\) is unique because it is the only value of \(x\) that satisfies the equation \(6ax + 2b = 0\). Since the second derivative changes sign at this point, it confirms that this is indeed an inflection point of the curve.

Key Concepts

Second Derivative

Understanding the second derivative is crucial in identifying inflection points on a curve. The second derivative, denoted as \(y''\), gives us information about the curvature of the graph of a function. It tells us how the rate of change of the slope itself changes. When the second derivative is positive, the function is concave up, meaning it forms a U-shape. Conversely, a negative second derivative indicates concave down, similar to an upside-down U-shape.
In the context of finding inflection points, we look for points where the second derivative equals zero and changes sign at that point. This is because the sign change indicates a shift from concave up to concave down, or vice versa, marking a transition in curvature. In our given cubic polynomial \(y=ax^3 + bx^2 + cx + d\), the second derivative is \(y'' = 6ax + 2b\). Solving \(6ax + 2b = 0\) provides the inflection point's \(x\)-coordinate.

Cubic Polynomial

A cubic polynomial is a polynomial of degree three, taking the form \(y = ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants, with \(a eq 0\). This type of polynomial can have up to three real roots and can exhibit up to two turning points and one inflection point.
The graph of a cubic polynomial has varying shapes depending on the coefficients, particularly \(a\). The sign of \(a\) indicates whether the cubic tends to rise or fall as \(x\) moves from negative to positive infinity.

• If \(a > 0\), the cubic polynomial typically begins in the lower left and rises to the upper right.
• If \(a < 0\), it does the opposite, starting from the upper left and falling to the lower right.

By calculating the first and second derivatives, we can analyze its critical points, providing insights into the curve's behavior.

Curve Analysis

Curve analysis involves examining the behavior of a curve concerning its slopes and curvatures, often through derivatives. Key aspects of curve analysis include finding:

• **Critical Points**: Locations on the graph where the first derivative is zero or undefined, possibly indicating a local maximum or minimum.
• **Inflection Points**: Points where the second derivative equals zero and changes sign, indicating a change in curvature.
• **Concavity**: Determined by the second derivative. Positive implies the graph is concave up; negative, concave down.

In our context, for the polynomial \(y = ax^3 + bx^2 + cx + d\), identifying these points helps in understanding the overall shape of the curve. The specific inflection point can be located by setting the second derivative to zero, then verifying sign changes, emphasizing versatility in interpreting complex graph behaviors.

Calculus Problem Solving

Using calculus to solve problems involving curves allows for precise understanding of their features. This typically involves the application of derivatives to find critical points like maximums, minimums, and inflection points, as seen with our exercise.
In addressing the current problem, we:

• Calculating the first derivative \(y' = 3ax^2 + 2bx + c\) to understand slope changes.
• Finding the second derivative \(y'' = 6ax + 2b\) for insights on curvature and to spot inflection points.

For a cubic polynomial, applying these calculus tools efficiently determines where the graph's nature shifts. Also, interpreting the results—in this case, confirming a unique inflection point at \(x = -\frac{b}{3a}\)—is crucial. This approach is broadly applicable, helping solve real-world problems involving dynamic rates of change.