Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

Consider a general quartic curve \(y=a x^{4}+b x^{3}+\) \(c x^{2}+d x+e\), where \(a \neq 0\). What is the maximum number of inflection points that such a curve can have?

Step-by-Step Solution

Answer

A quartic curve can have at most 2 inflection points.

Step 1: Understanding Inflection Points

An inflection point is where the curve changes concavity, which is determined by the second derivative. For a maximum number of inflection points, we need to find out when the second derivative equals zero and analyze the sign change.

Step 2: Compute the First Derivative

The first derivative of the function \[y = ax^4 + bx^3 + cx^2 + dx + e\] is \[\frac{dy}{dx} = 4ax^3 + 3bx^2 + 2cx + d.\]

Step 3: Compute the Second Derivative

The second derivative of the function is given by \[\frac{d^2y}{dx^2} = 12ax^2 + 6bx + 2c.\] We will use this equation to find inflection points.

Step 4: Solve for Inflection Points

To find potential inflection points, set the second derivative equal to zero and solve: \[12ax^2 + 6bx + 2c = 0.\] This is a quadratic equation in terms of \(x\).

Step 5: Determine Number of Solutions for Quadratic Equation

The solutions to the quadratic equation are found using the discriminant \(D = (6b)^2 - 4\cdot12a\cdot2c = 36b^2 - 96ac\). If \(D > 0\), there are 2 solutions; if \(D = 0\), there is 1 solution; if \(D < 0\), there are no solutions.

Step 6: Identify Maximum Inflection Points

Since the quadratic can have at most 2 distinct real solutions, the quartic curve can have at most 2 inflection points because each real root of the second derivative can correspond to one inflection point.

Key Concepts

Inflection Points

Inflection points are special places on a curve where the curve changes its concavity or bending direction. The concavity of the curve is determined by the nature of the second derivative. When a curve moves from concave up (like a cup) to concave down (like a cap), this transition occurs at an inflection point.

To find inflection points, we need to look at where the second derivative equals zero. But that's not all — the second derivative must also change signs at these points to confirm a true inflection point.

In our quartic curve, which is of the form \[y = ax^4 + bx^3 + cx^2 + dx + e\], we can have up to two inflection points. This is determined by solving the second derivative equation and checking for these zero-crossings that result in a change of sign.

Concavity

Concavity describes how a curve bends. When a curve is concave up, it looks like a cup holds water. Meanwhile, when it is concave down, it resembles a cap or umbrella. The concavity of a curve is dependent on the sign of its second derivative.

• When the second derivative is positive, the curve is concave up.
• When the second derivative is negative, the curve is concave down.

Understanding concavity is crucial for interpreting the behavior of graphs and functions, especially when it comes to identifying inflection points or determining intervals of increase and decrease in the curve's behavior.

Second Derivative

The second derivative of a function gives us essential information about the function's concavity. In calculus, the second derivative is the derivative of the derivative — it measures how the rate of change itself is changing.

For our quartic function, the second derivative is given by \[\frac{d^2y}{dx^2} = 12ax^2 + 6bx + 2c\]. We set this equal to zero to solve for possible inflection points.

The change in the sign of the second derivative from positive to negative (or vice versa) as we move across the graph tells us whether a curve moves from being concave up to concave down, thus pinpointing inflection points.

This change in sign can help ascertain the nature of stationary points as well.

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, typically in the form \[ax^2 + bx + c = 0\]. The solution to a quadratic equation can provide different outcomes based on the discriminant, which is the part of the quadratic formula under the square root \[(b^2 - 4ac)\].

For our second derivative \[12ax^2 + 6bx + 2c\], the quadratic nature emerges when we set it to zero to find potential inflection points. The discriminant here, \[D = 36b^2 - 96ac\], tells us the number of real solutions, which approximates to the number of inflection points.

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root.
• If \(D < 0\), no real roots exist.

Hence a quartic curve could potentially have at most two inflection points depending on these roots.