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

Quadratic curves have no inflection points as their second derivative is constant.

Step 1: Understanding Inflection Points

Inflection points occur where the second derivative of a function changes its sign, meaning the concavity changes from concave up to concave down or vice versa. To find these points, we need to analyze the second derivative of the quadratic equation.

Step 2: Find the First Derivative

The given quadratic function is \( y = ax^2 + bx + c \). To find the first derivative, we use the power rule: \( \frac{dy}{dx} = 2ax + b \).

Step 3: Find the Second Derivative

Differentiating the first derivative \( 2ax + b \) with respect to \( x \), we get the second derivative: \( \frac{d^2y}{dx^2} = 2a \).

Step 4: Analyzing the Second Derivative

The second derivative \( \frac{d^2y}{dx^2} = 2a \) is a constant, meaning it does not depend on \( x \). This implies that the concavity of the function does not change, as the sign of the second derivative is constant.

Step 5: Conclusion on Inflection Points

Since the second derivative is constant and does not vary with \( x \), the quadratic function \( y = ax^2 + bx + c \) has no points where the concavity changes. Therefore, there are no inflection points.

Key Concepts

Inflection Points

Inflection points in the graph of a function are places where the curve changes its concavity. This means the curve bends in a new direction – from curving upwards (concave up) to downwards (concave down), or the other way around. To find inflection points, we generally look at the second derivative of the function. The second derivative tells us about the concavity by indicating whether the curve is bending upwards or downwards.

For a function to have an inflection point, the second derivative must change sign from positive to negative or negative to positive. If the second derivative is always positive or always negative, it means the curve never changes its concavity. This is exactly the case with quadratic functions like the one given by the formula: - Quadratic Function: \( y = ax^2 + bx + c \) - First Derivative: \( rac{dy}{dx} = 2ax + b \) - Second Derivative: \( \frac{d^2y}{dx^2} = 2a \)
Since \( 2a \) is a constant and doesn’t depend on \( x \), our quadratic function has no inflection points. Its concavity stays the same everywhere on the graph.

Calculus

Calculus is like the toolset of advanced mathematics that helps us understand change and motion. It’s divided into two main parts:

• Differential Calculus: This focuses on the concept of a derivative, which relates to finding the rates of change, like the slope of a curve or the speed of an object.
• Integral Calculus: This focuses on integrals and gathering functions, which can be understood as finding areas under curves or the total accumulation of quantities.

When we talk about derivatives in calculus, we're referring to finding the slope or the rate of change at any given point on a graph. The tools of differential calculus are important when dealing with functions to understand their behavior, like determining where they might have inflection points or where they might change direction. In our quadratic equation exercise, you're using differential calculus to determine that there are no inflection points present by examining the second derivative.

Derivatives

Derivatives are one of the fundamental building blocks of calculus. They measure how a function changes as its inputs change. For example, if you have a function \( y=ax^2+bx+c \), taking the derivative gives you the rate at which \( y \) changes as \( x \) changes.

For our quadratic function, the first derivative calculated as \( \frac{dy}{dx} = 2ax + b \) shows the slope of the tangent line at any point \( x \). It's like zooming in on a small piece of the curve to see just how steep it is. When you differentiate this first derivative again with respect to \( x \), you get the second derivative \( \frac{d^2y}{dx^2} = 2a \).

This second derivative is crucial for understanding the concavity of the graph:

• If \( \frac{d^2y}{dx^2} \) is positive, the curve is concave up like a cup.
• If \( \frac{d^2y}{dx^2} \) is negative, it's concave down like a frown.
• If it's constant, as is the case here, this confirms that the curve has no inflection points.

In simple terms, the derivative gives us powerful insights into the behavior of functions by quantifying how they change and what kind of shape they’re taking on their graphs.