Problem 26 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ y=x^{3}-3 x+2 $$

Step-by-Step Solution

Answer

The critical numbers of the function \(y = x^{3} - 3x + 2\) are \(x = -1\) and \(x = 1\). The function is decreasing for \(x\) in the intervals \((-∞, -1)\) and \((1, ∞)\) while it's increasing for \(x\) in the interval \((-1, 1)\).

Step 1: Find the Derivative

The derivative of the function \(y = x^{3} - 3x + 2\) is obtained using the power rule for derivatives. In this case, the derivative, which we'll denote as \(y'\), is \(y' = 3x^{2} - 3\).

Step 2: Find Critical Numbers

The critical numbers are found by setting the derivative equal to zero and solving for \(x\). Thus we have, \(3x^{2} - 3 = 0\), simplifying gives \(x^{2} = 1\), and further solving yields two critical numbers \(x = -1\) and \(x = 1\).

Step 3: Apply First Derivative Test

To apply the First Derivative Test, we use our critical numbers to create intervals and find the sign of the derivative within those intervals. We take \(x < -1\), \(-1 < x < 1\), and \(x > 1\). After testing these intervals in the derivative, we find the sign of terms and consequently that the function \(y\) is decreasing over \((-∞, -1)\), increasing over \((-1, 1)\), and decreasing over \((1, ∞)\).

Step 4: Use Graphing Utility

Though we can't provide a graph here, on graphing using a graphing utility, the function \(y = x^{3} - 3x + 2\) will display the aforementioned intervals of increase and decrease, confirming our solution.

Key Concepts

First Derivative Test

The First Derivative Test is a crucial tool in calculus for determining where a function is increasing or decreasing, and for identifying local maxima and minima. When we find the derivative of a function and set it to zero, the values of x we get are known as 'critical numbers'. These numbers are potential candidates for local extrema.

To use the First Derivative Test, we first locate the critical numbers, then we examine the sign of the derivative to the left and right of each critical number. If the derivative changes from positive to negative, the function has a local maximum at that critical number. Conversely, if the derivative changes from negative to positive, there's a local minimum.

For our exercise, the critical numbers are \( x = -1 \) and \( x = 1 \). By testing intervals around these numbers, we observed the function's behavior—its tendency to increase or decrease—which is pivotal in sketching the graph of the function and understanding its overall behavior.

Increasing and Decreasing Functions

Understanding when a function is increasing or decreasing is fundamental in analyzing the shape of its graph. A function is said to be increasing on an interval if the output values get larger as the input values increase. Conversely, it is decreasing if the output gets smaller as the input increases.

To determine this, we observe the sign of the first derivative. If the first derivative is positive over an interval, it indicates that the function is increasing there. If the derivative is negative, the function is decreasing.

In our exercise example, after identifying the critical numbers \( x = -1 \) and \( x = 1 \) and testing the sign of the derivative in the intervals around them, it was concluded that the function is decreasing for \( x < -1 \) and \( x > 1 \) and increasing for \( -1 < x < 1 \). This information not only helps plot the graph accurately but also assists in predicting the behavior of the function.

Power Rule Derivatives

The Power Rule is one of the most applied rules when taking derivatives in calculus. It states that if you have a function \( f(x) = x^n \) where \( n \) is any real number, then the derivative of \( f \) with respect to \( x \) is \( f'(x) = nx^{n-1} \).

This rule significantly simplifies the calculation of derivatives for polynomial functions. When applied to our exercise \( y = x^3 - 3x + 2 \), we differentiate each term separately using the Power Rule. For the term \( x^3 \) the derivative is \( 3x^2 \) and for \( -3x \) it's \( -3 \) (considering \( x^1 \) where \( n=1 \), so \( nx^{n-1} = 1* x^{1-1} = 1*x^0 = 1 \)). There is no x in the constant term \( +2 \) so its derivative is zero.

The Power Rule is particularly efficient in identifying critical numbers, which are used in the First Derivative Test to define intervals of increasing or decreasing functions. As described, the derivative \( y' = 3x^2 - 3 \) was quickly found using the Power Rule, demonstrating its usefulness in solving calculus problems.