Problem 107 - Thomas Calculus in SI Units

Question

What can you say about the inflection points of a quadratic curve \(y=a x^{2}+b x+c, a \neq 0 ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

A quadratic curve has no inflection points because its second derivative is constant and non-zero.

Step 1: Understand the Definition of Inflection Points

An inflection point is where the curve changes concavity — from concave up (cup-shaped) to concave down (cap-shaped) or vice versa. To find an inflection point, we typically look for where the second derivative of a function equals zero and changes sign.

Step 2: Find the Second Derivative of the Quadratic Function

Given the function \[y = ax^2 + bx + c,\]the first derivative is \[y' = 2ax + b.\]The second derivative then becomes \[y'' = 2a.\]

Step 3: Analyze the Second Derivative

The second derivative, \(y'' = 2a\), is a constant. It does not depend on \(x\), meaning the concavity of the quadratic curve (determined by the sign of \(2a\)) does not change. Therefore, there is no point on the curve where the concavity changes.

Step 4: Conclusion About Inflection Points

Since the second derivative is constant and non-zero, the quadratic function does not have any inflection points. A quadratic function is either concave up everywhere if \(a > 0\) or concave down everywhere if \(a < 0\).

Key Concepts

Quadratic Curve

When we discuss quadratic curves, we refer to functions of the form \[ y = ax^2 + bx + c \] where \(a eq 0\). These curves are parabolic in shape. Because of the square term, the graph of a quadratic function forms a unique "U" shape, either opening upwards or downwards. This shape distinguishes quadratic curves from linear or cubic curves.

• If \(a > 0\), the parabola opens upwards (like a smile) - a shape called concave up.
• If \(a < 0\), the parabola opens downwards (like a frown) - a shape called concave down.

The vertex of the parabola is either the maximum point or minimum point, depending on the orientation of the parabola. This point is not an inflection point but rather the peak or the lowest point of the curve. Because quadratic curves are simple smooth graphs, they always maintain a consistent direction in concavity and do not exhibit a change in their curvature.

Second Derivative

To understand properties like concavity of a curve, we need to delve into derivatives. For a quadratic function \[ y = ax^2 + bx + c, \] we start by finding its derivative. The first derivative \[ y' = 2ax + b \] gives us the slope or the rate of change of the curve at any point.
Taking the derivative of \(y'\), we obtain the second derivative: \[ y'' = 2a. \] This calculation reveals that for a quadratic function, the second derivative is a constant. It does not depend on \(x\), unlike more complex functions.

• The value of \(2a\) indicates the consistent "direction" of the concavity.
• A constant second derivative means the curve does not have any flexibility or points of changing concavity.

This constancy directly leads to the understanding that a quadratic function does not have any inflection points.

Concavity

Concavity deals with the "bending" nature of the curve. We say a curve is concave up when it bends upwards like a cup opening to the sky. It's concave down when it arches downwards like an upside-down bowl. To assess where a curve changes from concave up to concave down, we rely on the second derivative.
Since for a quadratic function \[ y'' = 2a \] is non-zero and constant, the function maintains either concavity up or down across its entire domain. This behavior is predictable once we determine the sign of \(a\):

• If \(a > 0\), it is concave up everywhere – always forming a smile.
• If \(a < 0\), it is concave down everywhere – always forming a frown.

There are no inflection points in a quadratic curve because the change in concavity never occurs. The curve is uniform in its bending, a trait specific to quadratic functions.