Question

General quartic Show that the general quartic (fourth-degree) polynomial \(f(x)=x^{4}+a x^{3}+b x^{2}+c x+d\) has either zero or two inflection points, and that the latter case occurs provided that \(b<3 a^{2} / 8\)

Short answer

Question: Show that the given quartic polynomial \(f(x)=x^{4}+a x^{3}+b x^{2}+c x+d\) has either zero or two inflection points, depending on the value of b with respect to \(3a^2/8\). Answer: The quartic polynomial will have two inflection points if \(b ≤ \frac{3a^2}{8}\). If \(b > \frac{3a^2}{8}\), it will have zero inflection points.

Step-by-step solution

Find the first and second derivatives of the function

To find the inflection points of the function, we need to calculate the second derivative and solve the equation, where the second derivative is equal to zero. First, let's find the first and second derivatives: Given function: \(f(x)=x^{4}+a x^{3}+b x^{2}+c x+d\) First derivative: \(f'(x)= \frac{d}{dx}(x^{4}+a x^{3}+b x^{2}+c x+d) = 4x^3 + 3ax^2 + 2bx + c\) Second derivative: \(f''(x) = \frac{d^2}{dx^2}(4x^3 + 3ax^2 + 2bx + c) = 12x^2 + 6ax + 2b\)

Solve the equation for inflection points

To find the inflection points, we need to solve the equation \(f''(x) = 0\): \(12x^2 + 6ax + 2b = 0\) Now, let's analyze this quadratic equation in x.

Analyze the number of real roots of the quadratic equation

To determine the number of inflection points, we need to check the number of real roots of the quadratic equation using the discriminant Δ. The discriminant of the quadratic equation \(Ax^2 + Bx + C = 0\) can be given by: Δ = \(B^2 - 4AC\) Using the values in our equation: Δ = \((6a)^2 - 4(12)(2b) = 36a^2 - 96b\)

Check the conditions for zero or two inflection points

The quadratic equation has real roots (inflection points) if Δ ≥ 0. Thus, for zero or two inflection points, we have the following conditions: 1. For zero inflection points: Δ < 0, so \(36a^2 - 96b < 0\). If we rearrange it, we get \(b > \frac{3a^2}{8}\). 2. For two inflection points: Δ ≥ 0, so \(36a^2 - 96b ≥ 0\). If we rearrange it, we get \(b ≤ \frac{3a^2}{8}\). After performing this analysis, it can be concluded that the quartic equation will have two inflection points if \(b<3a^2/8\).

Key concepts

Quartic Polynomial

A quartic polynomial is a type of polynomial that has a degree of four, which means the highest exponent of the variable is four. These polynomials are in the form of f(x) = ax^4 + bx^3 + cx^2 + dx + e, where a, b, c, d, and e are coefficients, and a is not equal to zero. Quartic polynomials can have up to four roots and may display a range of interesting behaviors, such as having multiple maximums and minimums.

An example of a quartic polynomial is the function given in our exercise: f(x) = x^4 + ax^3 + bx^2 + cx + d. This function is a general representation, where a, b, c, and d can take any real number values. If we are to analyze the curvature of this polynomial, we look for its inflection points which signify a change in the concavity of the graph of the function. It is these points that contribute to the characteristic 'w' shape often associated with quartic polynomials.

Understanding the nature of these inflection points gives insight into the geometry of the graph. In the exercise, we show that a general quartic polynomial can have zero or two inflection points, which is determined by the coefficients of the polynomial and clarified through calculus techniques.

Second Derivative Test

The second derivative test is a valuable calculus tool that helps determine the concavity of a function and identify its inflection points. An inflection point is where the graph of the function changes concavity, such as from concave up to concave down or vice versa. In essence, at an inflection point, the curvature of the graph changes direction.

The process of using the second derivative test involves taking the derivative of the function twice. The first derivative, f'(x), represents the slope of the tangent line to the graph at any given point, while the second derivative, f''(x), tells us about the function's concavity. If f''(x) > 0, the function is said to be concave up, and if f''(x) < 0, it is concave down.

For our quartic polynomial f(x) = x^4 + ax^3 + bx^2 + cx + d, we calculated the second derivative and set it equal to zero to find potential inflection points. By solving the resulting quadratic equation obtained from f''(x) = 0, we can determine the number and location of inflection points. This step is crucial as it directly informs the shape of the graph and the behavior of the polynomial function.

Discriminant of a Quadratic Equation

When solving quadratic equations, understanding the discriminant is crucial. The discriminant provides vital information on the nature of the roots of the quadratic equation without actually calculating them. For an equation of the form Ax^2 + Bx + C = 0, the discriminant Δ is defined as Δ = B^2 - 4AC.

The value of the discriminant determines the number and type of solutions:

• If Δ > 0, there are two distinct real roots.
• If Δ = 0, there is exactly one real root (a repeated root).
• If Δ < 0, there are no real roots, which means the roots are complex.

In the context of our exercise, we analyze a quadratic equation obtained from the second derivative of a quartic polynomial to find its inflection points. By examining the discriminant of this quadratic equation, we can conclude whether the quartic polynomial has zero, one, or two inflection points without computing the actual roots. The relationship Δ ≥ 0 tells us that the polynomial has two inflection points provided that b ≤ 3a^2/8, and this can significantly affect the graph's shape. This relation between the discriminant and the graph's concavity illustrates the interconnectivity of algebra and calculus in analyzing polynomial functions.