Question

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

Short answer

Answer: A quartic polynomial of the given form has either zero or two inflection points. There will be two inflection points provided \(b < \frac{3a^2}{8}\).

Step-by-step solution

Compute the first derivative

Begin by taking the derivative of the given quartic polynomial: \(f'(x) = \frac{d}{dx}(x^{4} + ax^{3} + bx^{2} + cx + d)\) Using the power rule for differentiation, we get: \(f'(x) = 4x^3 + 3ax^2 + 2bx + c\)

Compute the second derivative

Now, taking the derivative of the first derivative to obtain the second derivative: \(f''(x) = \frac{d}{dx}(4x^3 + 3ax^2 + 2bx + c)\) Again, using the power rule for differentiation, we get: \(f''(x) = 12x^2 + 6ax + 2b\)

Solve the second derivative for zero

In order to find the inflection points, we need to solve the second derivative for zero: \(12x^2 + 6ax + 2b = 0\) This is a quadratic equation in x, so we can apply the quadratic formula to find its roots: \(x = \frac{-6a \pm \sqrt{(-6a)^2 - 4(12)(2b)}}{2(12)}\)

Check the discriminant

Now, we need to analyze the discriminant of the quadratic equation, which is the part inside the square root in the quadratic formula: \(\Delta = (-6a)^2 - 4(12)(2b) = 36a^2 - 96b\) If \(\Delta < 0\), the second derivative equation has no real solutions, so there are no inflection points. If \(\Delta = 0\), the second derivative equation has one real solution. However, this is a repeated root, so the second derivative does not change sign across this point, and therefore, the graph will not change concavity; there will be zero inflection points in this case. If \(\Delta > 0\), the second derivative equation has two real solutions, and there will be two inflection points.

Determine the condition for existence of inflection points

So, we want to find when \(\Delta\) is positive: \(36a^2 - 96b > 0\) Divide both sides by 12: \(3a^2 - 8b > 0\) Add \(8b\) to both sides: \(3a^2 > 8b\) Divide by 3: \(a^2 > \frac{8}{3}b\) In conclusion, a fourth-degree polynomial with the given form has either zero or two inflection points, and there will be two inflection points provided \(b < \frac{3a^2}{8}\).

Key concepts

Quartic Polynomial

A quartic polynomial is a polynomial of degree four, and it takes the general form: \[ f(x) = x^4 + ax^3 + bx^2 + cx + d \] where \( a \), \( b \), \( c \), and \( d \) are coefficients that dictate the specific shape of the polynomial.
The highest degree term \( x^4 \) determines that it is indeed cubic in shape, and this gives the graph a characteristic shape.
Quartic polynomials can have up to four roots, three turning points, and in some cases, it can exhibit complex behavior like having multiple changes in direction.
These properties make quartic polynomials particularly interesting and sometimes challenging when analyzing their graphs.

Second Derivative

The second derivative of a function provides information about the concavity of the graph.
It is found by differentiating the first derivative of the function.
For the quartic polynomial \( f(x) = x^4 + ax^3 + bx^2 + cx + d \), the first derivative is:
\[ f'(x) = 4x^3 + 3ax^2 + 2bx + c \] To find the second derivative, you differentiate the first derivative, resulting in:
\[ f''(x) = 12x^2 + 6ax + 2b \] By setting \( f''(x) = 0 \), you can find potential inflection points, which are points where the concavity changes.

Discriminant

The discriminant in the context of quadratic equations helps to determine the nature of the roots.
For the second derivative \( f''(x) = 12x^2 + 6ax + 2b \), the discriminant \( \Delta \) is:
\[ \Delta = (-6a)^2 - 4(12)(2b) = 36a^2 - 96b \]

• If \( \Delta > 0 \), there are two distinct real roots, indicating two inflection points exist.
• If \( \Delta = 0 \), there is one repeated real root, so no inflection points occur.
• If \( \Delta < 0 \), there are no real roots, hence, no inflection points.

This analysis aids in understanding when and how a quartic polynomial changes its concavity, contributing to the graph's overall shape.

Concavity

Concavity of a function describes the direction in which it curves.
A graph is concave up if it curves upwards like a cup, and concave down if it curves downwards.
The second derivative helps determine the concavity:

• If \( f''(x) > 0 \), the graph is concave up at that interval.
• If \( f''(x) < 0 \), the graph is concave down.
• If \( f''(x) = 0 \), this indicates a possible inflection point, where the graph could change from concave up to concave down, or vice versa.

Understanding concavity is essential for sketching graphs accurately and for solving optimization problems, as it provides valuable insights into the behavior of the function.