Problem 62 - Algebra and Trigonometry

Question

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=6 x^{3}+3 x+1 $$

Step-by-Step Solution

Answer

The polynomial has no local maxima or minima.

Step 1: Understanding the Polynomial

The function we need to graph is \( y = 6x^3 + 3x + 1 \). This is a cubic polynomial, which typically has one or two stationary points where local maxima or minima may occur.

Step 2: Finding the Derivative

To find the local maxima and minima, we first compute the derivative of the polynomial. Calculating the derivative gives us \( y' = \frac{d}{dx}(6x^3 + 3x + 1) = 18x^2 + 3 \).

Step 3: Solving for Critical Points

Set the derivative equal to zero to find the critical points: \( 18x^2 + 3 = 0 \). Solving for \( x \), we get \( 18x^2 = -3 \), which means \( x^2 = -\frac{1}{6} \). However, \( x^2 = -\frac{1}{6} \) has no real solutions as the square of a real number cannot be negative.

Step 4: Analyzing the Findings

Since the equation \( 18x^2 + 3 = 0 \) has no real solutions, there are no critical points for this function where the derivative is zero. Therefore, there are no local maxima or minima.

Step 5: Conclusion

As there are no critical points with the derivative equal to zero, the cubic function \( y = 6x^3 + 3x + 1 \) has no local maxima or minima. The graph of the polynomial will display end behaviors typical to cubic functions: rising to the right and falling to the left or vice versa, but with no peaks or troughs in between.

Key Concepts

Derivative

A derivative represents the rate at which a function changes at any given point. For a polynomial, taking the derivative involves differentiating each term according to the power rule. In this exercise, the polynomial is given as \( y = 6x^3 + 3x + 1 \). The derivative of this function, denoted as \( y' \), is derived by applying the power rule to each individual term.

• The derivative of \( 6x^3 \) is \( 18x^2 \), since you multiply by the exponent and reduce it by one.
• The derivative of \( 3x \) is \( 3 \), as the exponent 1 reduces to zero which eliminates the \( x \) term.
• The derivative of a constant, such as \( 1 \), is \( 0 \), because constants do not change.

Thus, our derivative is \( y' = 18x^2 + 3 \). Calculating the derivative helps us find where a function might stop increasing or decreasing, these points are called critical points.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points can indicate where a function changes direction, either from increasing to decreasing or vice versa. To find the critical points of our cubic polynomial, we solve the equation \( 18x^2 + 3 = 0 \).

• First, isolate the quadratic term: \( 18x^2 = -3 \).
• Then, divide each side by 18 to simplify: \( x^2 = -\frac{1}{6} \).

Here we encounter a problem. The equation \( x^2 = -\frac{1}{6} \) is impossible to solve within the realm of real numbers since a square of a real number cannot be negative. Therefore, there are no real critical points for the function \( y = 6x^3 + 3x + 1 \), meaning it does not change direction anywhere on the real number line.

Local Maxima and Minima

Local maxima and minima are the peaks and troughs in a graph, where a function reaches a highest or lowest value in a particular interval. To identify these points, we typically use critical points. However, in this case, since the derivative \( 18x^2 + 3 \) has no real solutions, it indicates that there are no real critical points at which the function changes direction.

Thus, the cubic function \( y = 6x^3 + 3x + 1 \) doesn't have any local maxima or minima. This means the graph of the function will continually increase or decrease without any turning points, preventing it from having any peaks (local maxima) or valleys (local minima). Understanding this helps in predicting the overall shape and behavior of the graph.

Cubic Function Behavior

Cubic polynomial functions, such as \( y = 6x^3 + 3x + 1 \), have a distinct behavior based on their structure. Typically, they can have up to two turning points where they achieve local maxima or minima. However, since our function doesn’t have real solutions for its derivative, it doesn't exhibit typical turning points.

The behavior of cubic functions on a graph often includes:

• Monotonically increasing or decreasing without interruption when there are no critical points.
• Its end behavior includes rising to the right and falling to the left or vice versa, a signature trait of functions led by a cubic term.

Despite the lack of local maxima or minima, understanding the end behavior helps visualize the overall trajectory of the graph. Such knowledge is crucial when analyzing or graphing polynomial functions, as it allows us to predict how the function progresses through the coordinate plane.