Problem 67 - Algebra and Trigonometry

Question

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=x^{8}-3 x^{4}+x $$

Step-by-Step Solution

Answer

The polynomial has several turning points, indicating multiple local maxima and minima.

Step 1: Identify the polynomial function

The given polynomial is \( y = x^8 - 3x^4 + x \). This is an eighth-degree polynomial, meaning it could have up to 7 critical points, and therefore up to 6 local maxima or minima.

Step 2: Find the first derivative

To find the local maxima and minima of a function, we first need to calculate its first derivative. The derivative of \( y = x^8 - 3x^4 + x \) is: \[ y' = 8x^7 - 12x^3 + 1 \].

Step 3: Set the derivative to zero and solve for critical points

Set \( y' = 8x^7 - 12x^3 + 1 = 0 \) and solve for \( x \) to find the critical points. Solving this equation requires numerical methods or graphing tools, as it does not have simple algebraic roots.

Step 4: Analyze critical points using the second derivative

Calculate the second derivative \( y'' \) to determine the nature of each critical point. The second derivative is \( y'' = 56x^6 - 36x^2 \). Use the sign of \( y'' \) at each critical point to determine if it is a local maximum (\( y'' < 0 \)) or a local minimum (\( y'' > 0 \)).

Step 5: Graph the polynomial

Graph the function \( y = x^8 - 3x^4 + x \) using graphing software or a calculator. Identify the turning points to verify the number of local maxima and minima visually.

Step 6: Count the local maxima and minima

From graphing the polynomial and analyzing the sign of the second derivative at critical points, determine the exact number of local maxima and minima.

Key Concepts

Local Maxima

In the study of polynomial functions, understanding local maxima is essential for analyzing their behavior. A local maximum refers to a point on a graph where the function changes direction from increasing to decreasing. This means the function reaches a peak at this point, with no higher points immediately adjacent.
To find local maxima, you first identify critical points by setting the first derivative of the function to zero. However, to distinguish whether these points are maxima or minima, the second derivative test is used.

• Calculate the second derivative.
• At each critical point, plug the values into the second derivative.
• If the result is negative ( ( y'' < 0 )), the critical point is a local maximum.

Remember, the second derivative provides insight into the concavity of the graph, thus helping confirm the nature of the turning point.

Local Minima

Local minima serve as the opposite concept to maxima, signifying points where the function shifts from decreasing to increasing, showing the lowest point locally in its vicinity. These points are quite crucial in depicting the valleys in the graph of a polynomial function.
The process of finding local minima also involves analyzing critical points discovered via the first derivative.

• Once these critical points are identified, use the second derivative test.
• If the second derivative is positive ( (y'' > 0) ) at a critical point, then the function has a local minimum there.

This helps ensure that the critical points not mistaken as minima when they are not. It stands as valuable for understanding the behavior of polynomial functions across different intervals.

First Derivative

The first derivative of a polynomial function is a key tool in calculus. It represents the rate of change or the slope of the function at any given point. By setting the first derivative to zero, you can determine where the function’s slope is zero, indicating potential local maxima or minima.
For the function \( y = x^8 - 3x^4 + x \), the first derivative is found as follows: \[ y' = 8x^7 - 12x^3 + 1 \]
This derivative is crucial as it helps identify critical points, where the slope of the tangent is zero. These critical points are calculated by solving \( y' = 0 \), and although it might require numerical solutions, it lays the foundation for evaluating maxima and minima.
The first derivative provides insight into where the turning points of the polynomial lie, making it essential for graph interpretation.

Second Derivative

The second derivative represents the rate of change of the first derivative and is instrumental in determining the concavity of the function's graph. When we have the critical points from the first derivative, the second derivative supplies the necessary data to verify the nature of these points.
For the function \( y = x^8 - 3x^4 + x \), the second derivative is given as:\[ y'' = 56x^6 - 36x^2 \]
This helps analysts determine:

• If \( y'' < 0 \), the function is concave down, indicating a local maximum.
• If \( y'' > 0 \), the function is concave up, signaling a local minimum.

The second derivative test is a vital step in understanding the behavior and the shape of the polynomial function. It ensures an accurate interpretation of the function’s graphical features and their implications.