Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Consider a general quadratic curve \(y=a x^{2}+b x+c\). Show that such a curve has no inflection points.

Step-by-Step Solution

Answer

A quadratic curve has no inflection points because its second derivative is constant.

Step 1: Differentiating the Curve

To determine inflection points, we first need to calculate the first derivative of the given quadratic curve. The curve is given by the equation \(y = ax^2 + bx + c\). The first derivative of this equation with respect to \(x\) is calculated as follows: \(\frac{dy}{dx} = 2ax + b\).

Step 2: Calculate the Second Derivative

Inflection points occur where the concavity changes, which is determined using the second derivative. We differentiate the first derivative \(\frac{dy}{dx} = 2ax + b\) again with respect to \(x\) to find the second derivative: \(\frac{d^2y}{dx^2} = 2a\).

Step 3: Analyze the Second Derivative

Observe the second derivative \(\frac{d^2y}{dx^2} = 2a\). This is a constant value, meaning it does not depend on \(x\). For a point to be an inflection point, the second derivative must change signs, indicating a change in concavity. However, since \(2a\) is constant, it never changes signs.

Step 4: Conclusion on Inflection Points

Since the second derivative \(\frac{d^2y}{dx^2} = 2a\) is constant, the concavity of the curve does not change. Therefore, the quadratic curve \(y = ax^2 + bx + c\) has no points where the concavity changes, indicating that it has no inflection points.

Key Concepts

Quadratic Curves

quadratic curve is a type of polynomial equation of the second degree. These equations are expressed in the standard form of \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These curves are characterized by their parabolic shape.

The graph of a quadratic curve is a parabola which can open upwards or downwards, determined by the sign of the constant \(a\). If \(a > 0\), the parabola opens upwards like a U-shape, and if \(a < 0\), it opens downwards like an inverted U.

Quadratic curves are symmetric about a vertical line called the axis of symmetry. This symmetry line passes through the vertex of the parabola, which is the highest or lowest point on the curve, depending on the direction the parabola opens.

• The vertex can be found using the formula \(x = -\frac{b}{2a}\) to get the x-coordinate. Substituting this back into the original equation provides the y-coordinate.

Understanding quadratic curves is crucial because they represent many real-world phenomena where relationships between variables involve squared terms.

Second Derivative

The second derivative of a function provides insights into the function's concavity and, consequently, its inflection points. By deriving the first derivative of a function and then differentiating it once more, we arrive at the second derivative.

In the given quadratic curve \(y = ax^2 + bx + c\), the first derivative is \(\frac{dy}{dx} = 2ax + b\). Differentiating this result one more time with respect to \(x\) gives us the second derivative: \(\frac{d^2y}{dx^2} = 2a\).

Interestingly, the second derivative of a quadratic function is always a constant because it only depends on the "a" term of the original function. This constant value, \(2a\), reveals that the rate of change of slope is uniform across the curve. Therefore, there can be no change in concavity, which is essential for the presence of inflection points.

Concavity

Concavity refers to the direction in which a curve bends. It is an essential concept when analyzing the behavior of curves, especially for understanding the shape and turning points of graphs.

With quadratic curves, the second derivative tells us a lot about concavity:

• If the second derivative is positive (\(\frac{d^2y}{dx^2} > 0\)), the curve is concave up, meaning it bends upward like a U.
• If the second derivative is negative (\(\frac{d^2y}{dx^2} < 0\)), the curve is concave down, like an upside-down U.

For a curve to have an inflection point, its concavity must change from up to down or vice versa. However, since the second derivative of any quadratic curve is constant, the concavity does not change. This is why quadratic curves do not feature inflection points, as the uniform concavity is maintained throughout the entire curve. Understanding concavity and how it's depicted by the second derivative helps in graphically interpreting and predicting the behavior of polynomial functions.