Problem 60 - Precalculus Mathematics for Calculus

Question

Graph the polynomial and determine how many local maxima and minima it has. $$y=x^{3}+12 x$$

Step-by-Step Solution

Answer

The polynomial has no local maxima or minima.

Step 1: Identify the Polynomial

The given polynomial is \( y = x^3 + 12x \). This is a cubic polynomial, which typically has an S-shaped curve when graphed.

Step 2: Compute the Derivative

To find the local maxima and minima, we first need the derivative of \( y \). The derivative of \( y = x^3 + 12x \) is \( y' = 3x^2 + 12 \).

Step 3: Find Critical Points

Set the first derivative equal to zero to find critical points: \( 3x^2 + 12 = 0 \). Simplifying gives \( x^2 = -4 \). This equation has no real solutions, indicating no critical points where the slope is zero.

Step 4: Analyze the Second Derivative

Find the second derivative to analyze the concavity of the function. The second derivative is \( y'' = 6x \). Since there are no real solutions for the first derivative being zero, this step confirms the absence of local maxima or minima.

Step 5: Graph the Polynomial

Graphing \( y = x^3 + 12x \) shows the general shape of the function. The cubic term \( x^3 \) means the graph steadily increases or decreases and does not have local maxima or minima.

Key Concepts

Cubic Polynomial

A cubic polynomial is an algebraic expression of the form \(ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\). These polynomials are called "cubic" because the highest power of the variable \(x\) is three. A distinguishing feature of cubic polynomials is their unique shape when graphed, commonly known as an S-shaped curve. This is because the slope of the curve can change multiple times, reflecting the polynomial’s capacity to have turning points at local maxima and minima. However, not all cubic polynomials will exhibit these features, depending on the roots and coefficients of the equation.

In the exercise we analyzed, the polynomial \(y = x^3 + 12x\) doesn't have real solutions to make turning points through critical points, which means it continually increases or decreases without stabilizing around a horizontal tangent.

Derivatives in Calculus

Derivatives in calculus are tools that help to understand how a function changes at any point. The derivative represents the rate of change or the slope of the function's graph at a particular point. When you're looking at a polynomial like \(y = x^3 + 12x\), taking its derivative is crucial to studying its behavior.

For our exercise, the first derivative, \(y' = 3x^2 + 12\), reflects how the slope changes across different values of \(x\). By setting the first derivative to zero \(3x^2 + 12 = 0\), we attempt to locate potential points of local maximum or minimum, but discover that it provides no real solutions, indicating that there are no stationary points where the slope is zero. This means the function does not change direction in a typical local maximum or minimum way. Continuing on to the second derivative, \(y'' = 6x\), helps us determine the concavity of the function but, in this case, serves to confirm the lack of any local extrema as there's no zero point to analyze further.

Local Maxima and Minima

Local maxima and minima refer to the high and low turning points on a graph within a particular region. These are points where the function momentarily stops increasing or decreasing and reverses direction, defined by where the first derivative changes sign.

In calculus, specifically for polynomial functions, finding these points involves determining where the first derivative equals zero, finding critical points, and then further analyzing these with the second derivative test to confirm whether they are maxima or minima. In the exercise, because \(3x^2 + 12 = 0\) yields no real answers, there are no critical points, which implies that there are no local maxima or minima for \(y = x^3 + 12x\). Graphing also helps visualize this by showing a smooth, consistent increase or decrease without such directional changes, aligning with our derivative findings.