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?

Short answer

A quartic curve can have a maximum of 2 inflection points.

Step-by-step solution

Understanding Inflection Points

An inflection point on a curve is a point where the curvature changes sign. For a function to have an inflection point, its second derivative must change sign at that point.

Find the Second Derivative

Consider the function given: \(y = ax^4 + bx^3 + cx^2 + dx + e\). First, find the first derivative: \(y' = 4ax^3 + 3bx^2 + 2cx + d\). Now find the second derivative: \(y'' = 12ax^2 + 6bx + 2c\).

Setting the Second Derivative to Zero

To find potential inflection points, solve \(12ax^2 + 6bx + 2c = 0\). This is a quadratic equation in \(x\).

Potential Solutions of the Quadratic Equation

The quadratic \(12ax^2 + 6bx + 2c = 0\) may have 0, 1, or 2 real roots. The number of solutions corresponds to the number of potential points where the inflection occurs, depending on whether the sign changes at these points.

Determine Maximum Number of Inflection Points

A quartic curve can have a maximum of 2 real inflection points if the quadratic equation has 2 distinct real roots, and the second derivative changes sign at each of these roots.

Key concepts

Inflection Points

In the context of a curve, inflection points are critical moments where the curve changes its direction in terms of concavity. To determine these points, it means identifying where the curve shifts from being concave upwards (smile shape) to concave downwards (frown shape), or vice versa.

An inflection point is interesting because it signifies a change in the curvature of the graph. For a curve to possess an inflection point, the second derivative must not only change sign but must be equal to zero at that point. Finding inflection points is crucial for understanding the behavior of a polynomial, such as a quartic curve, which can reveal points of special interest like turning points or saddle points.

Understanding and identifying inflection points can provide insights into the overall shape and behavior of the polynomial curve you are studying.

Second Derivative

The second derivative of a function gives us information about the curvature of its graph. It is the derivation of the first derivative and helps in understanding how the rate of change itself is changing. In the case of the quartic curve, the second derivative is calculated by successively differentiating the original quartic function twice.

For our quartic curve, given the function \[ y = ax^4 + bx^3 + cx^2 + dx + e \] we first find the first derivative as:\[ y' = 4ax^3 + 3bx^2 + 2cx + d \].

The second derivative, therefore, becomes \( y'' = 12ax^2 + 6bx + 2c \).
This specific expression allows us to explore the conditions under which the curve transitions between different concavities, thus indicating possible inflection points through its sign changes.

Cubic Derivative

A cubic derivative, particularly in the form of the first derivative in our context, represents the slope function of the original quartic curve. By differentiating the quartic equation, we obtain a cubic expression that tells us where the curve is increasing or decreasing.

The cubic derivative for our quartic equation comes as:\[ y' = 4ax^3 + 3bx^2 + 2cx + d \].This represents a polynomial of degree three and can have up to three real roots. These roots indicate critical points, which can be turning points or part of the solutions for finding changes in concavity.

When analyzing the cubic derivative in conjunction with the second derivative, valuable interaction insights become obvious, like predicting where inflection points or alternative significant features of the curve could be.

Roots of Quadratic Equations

Finding the roots of a quadratic equation is fundamental in solving for inflection points in a quartic curve. The quadratic formed by the second derivative: \( 12ax^2 + 6bx + 2c = 0 \), must be solved to find potential inflection points of the quartic function.

The nature of the roots dictates the potential for inflection points:

• If it has two distinct real roots, it implies two potential inflection points.
• If it has one real root (a repeated root), it denotes one potential inflection point.
• If it has no real roots, then the quartic curve has no inflection points.

Solving the quadratic equation involves using the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \], where you determine the discriminant \(b^2 - 4ac\) to ascertain the number of roots.

This knowledge gives clarity on the amount and potential position of inflection points for the quartic curve under examination.