Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove that a quadratic function has no point of inflection.

Step-by-Step Solution

Answer

A quadratic function has no inflection points as its second derivative is constant.

Step 1: Define a Quadratic Function

A quadratic function is typically given in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). Our goal is to determine whether this function has a point of inflection.

Step 2: Calculate the First Derivative

To find points of inflection, we need the first derivative. The first derivative of the quadratic function \( f(x) = ax^2 + bx + c \) is \( f'(x) = 2ax + b \).

Step 3: Calculate the Second Derivative

For points of inflection, examine the second derivative. The second derivative of \( f(x) \) is \( f''(x) = 2a \).

Step 4: Analyze the Second Derivative

A point of inflection occurs where the second derivative changes sign, leading to \( f''(x) = 0 \) at that point. However, the second derivative \( f''(x) = 2a \) is constant (since \( a \) is a constant and \( a eq 0 \)), and therefore, it does not change sign.

Step 5: Conclude the Absence of Inflection Points

Since the second derivative \( f''(x) = 2a \) does not depend on \( x \) and does not change sign, the quadratic function \( f(x) = ax^2 + bx + c \) has no point of inflection.

Key Concepts

Point of Inflection

A point of inflection is a point on a curve where the curvature changes. This means that the graph of the function switches from being concave (curved upwards) to convex (curved downwards), or vice versa. For a point of inflection to exist, it typically requires the curve's second derivative to change signs at that point. In other words, on one side of the inflection point, the second derivative has a different sign than on the other side. This change indicates a shift in the direction of curvature.

However, a quadratic function such as \( f(x) = ax^2 + bx + c \) does not have any points of inflection. This is because its second derivative is a constant value, which we'll explore further below.

First Derivative

The first derivative of a function provides us with critical information about the function's slope, or rate of change, at any point \( x \). When we calculate the first derivative of a quadratic function \( f(x) = ax^2 + bx + c \), we get \( f'(x) = 2ax + b \).

This expression is linear and gives the slope of the tangent line to the curve at any \( x \). Understanding the first derivative helps us identify where the function is increasing or decreasing.

• If \( f'(x) > 0 \), the function is increasing.
• If \( f'(x) < 0 \), the function is decreasing.

For a point of inflection, while the first derivative is involved, it is primarily the second derivative’s behavior that determines its presence, as it describes how the slope itself changes.

Second Derivative

The second derivative of a function is crucial in understanding the curvature of the graph. It tells us how the rate of change of the function's slope (found with the first derivative) changes as \( x \) changes. For the quadratic function \( f(x) = ax^2 + bx + c \), its second derivative is \( f''(x) = 2a \).

This result is constant, as it depends only on \( a \) (a fixed number) and not \( x \). Since the second derivative does not change with \( x \), it cannot switch signs; therefore, quadratic functions do not have points of inflection. The second derivative being positive means the graph is always concave up, while a negative second derivative means it is always concave down, but it never changes between these states for a quadratic function.

Constant Function

A constant function is a type of function where the output value remains the same, no matter what the input \( x \) is. The formula can be written as \( f(x) = c \), where \( c \) is a fixed number. In this context, the term "constant function" refers to the characteristic of the second derivative of a quadratic function, \( f''(x) = 2a \).

This is considered a constant because its value does not change with different inputs of \( x \). It remains \( 2a \) for all \( x \), reflecting no variation, and therefore, no point at which the concavity can change. That's why, in a quadratic function, the fact that the second derivative is constant means that it doesn’t allow for a transition from concave to convex or vice versa, confirming the absence of points of inflection.