Problem 60 - Essential Calculus Early Transcendentals

Question

For what values of \(c\) does the polynomial \(P(x)=x^{4}+c x^{3}+x^{2}\) have two inflection points? One inflection point? None? Illustrate by graphing \(P\) for several values of \(c .\) How does the graph change as \(c\) decreases?

Step-by-Step Solution

Answer

Two inflection points for \(c > \sqrt{\frac{8}{3}}\) or \(c < -\sqrt{\frac{8}{3}}\), one for \(c = \pm \sqrt{\frac{8}{3}}\), none in between.

Step 1: Understand the Role of Inflection Points

Inflection points occur where the second derivative of the function changes from positive to negative or vice versa. In other words, it is where the concavity of the graph changes.

Step 2: Find the Second Derivative

First, find the first derivative of the polynomial: \( P'(x) = 4x^3 + 3cx^2 + 2x \). Then, find the second derivative: \( P''(x) = 12x^2 + 6cx + 2 \).

Step 3: Set the Second Derivative Equal to Zero

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

Step 4: Analyze the Discriminant

The nature of a quadratic equation is determined by its discriminant \( \Delta = b^2 - 4ac \), where \(a=12\), \(b=6c\), and \(c=2\). The discriminant is \( \Delta = (6c)^2 - 4(12)(2) = 36c^2 - 96 \).

Step 5: Determine the Number of Inflection Points

Analyze \( \Delta \) to determine the number of inflection points:- If \( \Delta > 0 \), the equation has two distinct real solutions, indicating two inflection points.- If \( \Delta = 0 \), the equation has one real solution, indicating one inflection point.- If \( \Delta < 0 \), the equation has no real solutions, indicating no inflection points.

Step 6: Solve for Specific Values of \(c\)

Solving \( 36c^2 - 96 = 0 \) gives the critical values for \(c\). Simplifying gives \( c^2 = \frac{96}{36} = \frac{8}{3} \), so \( c = \pm \sqrt{\frac{8}{3}} \). These are the boundary values between which the concavity changes.

Step 7: Interpret the Findings

- For \( c > \sqrt{\frac{8}{3}} \) or \( c < -\sqrt{\frac{8}{3}} \), \( \Delta > 0 \), there are two inflection points.- For \( c = \pm \sqrt{\frac{8}{3}} \), \( \Delta = 0 \), there is one inflection point.- For \( -\sqrt{\frac{8}{3}} < c < \sqrt{\frac{8}{3}} \), \( \Delta < 0 \), there are no inflection points.

Key Concepts

Second Derivative Test

The second derivative test is a helpful tool in calculus that can tell you a lot about the shape of a graph. It helps identify where a curve is concave up or concave down, and by extension, points of inflection, where the concavity changes.

For a given function, you first need to find the first derivative, which gives you the slope of the tangent line at any point. Then, you find the second derivative, which tells you how that slope is changing. When the second derivative is positive, the function is concave up, resembling a cup that holds water. Conversely, if the second derivative is negative, the function is concave down, like an upside-down cup.

Inflection points are the special points on the graph where the concavity changes from up to down or vice versa. To find these, set the second derivative to zero and solve for the variable. The solutions indicate potential inflection points, granted the concavity indeed changes at these solutions.

Concavity

Concavity is all about understanding whether a graph opens upwards or downwards. This concept is crucial when analyzing a function's behavior because it tells you how the graph is bending.

When we say a function is concave up, it bends upwards like a smile. Mathematically, this happens when the second derivative of a function, say \( f''(x) \), is greater than zero \( f''(x) > 0 \). If the graph is concave down, it resembles a frown, and the second derivative is less than zero \( f''(x) < 0 \).

Changing between these two states is what defines an inflection point—the spot where the graph changes direction in terms of its curvature. Understanding concavity gives you insights into how a function's values are increasing or decreasing, and how fast these changes are happening.

Discriminant Analysis

In solving polynomial equations, especially quadratics, discriminant analysis plays a significant role. The discriminant of a quadratic equation, viewed in the form \( ax^2 + bx + c = 0 \), is given by \( \Delta = b^2 - 4ac \).

This value tells us about the nature of the roots of the quadratic equation:

• If \( \Delta > 0 \), the quadratic has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root—a double root.
• If \( \Delta < 0 \), there are no real roots, only complex ones.

In the context of finding inflection points for polynomials like \( P(x) = x^4 + cx^3 + x^2 \), analyzing the discriminant of its second derivative helps determine how many times the concavity of the graph changes, which translates to how many inflection points the polynomial has.

Quadratic Equation Solutions

Solving quadratic equations is a fundamental part of calculus and algebra. A standard quadratic equation appears as \( ax^2 + bx + c = 0 \). To find the solutions of this equation, we use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, the expression under the square root, \( b^2 - 4ac \), is the discriminant, which determines the nature of the solutions.

The quadratic formula provides two solutions because of the \( \pm \) sign, corresponding to the possible roots of the equation. These roots are particularly useful when looking for zeros of a function or in cases like finding inflection points by setting the second derivative of a polynomial to zero.