Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Describe the increasing and decreasing behavior of the function. Find the point or points where the behavior of the function changes. \(f(x)=x^{2}-2 x\)

Step-by-Step Solution

Answer

The function \(f(x)=x^{2}-2 x\) is decreasing on the interval \(-( \infty,1)\) and increasing on the interval \((1, \infty)\). The behavior of the function changes at \(x=1\).

Step 1: Differentiate the Function

The first step is to differentiate the given function. The derivative of \(f(x)=x^{2}-2 x\) is \(f'(x) = 2x - 2\).

Step 2: Find Critical Points

Next, set the derivative equal to zero to find the critical points. Solving the equation \(2x - 2 = 0\) for \(x\) gives \(x = 1\).

Step 3: Analyze Function Behavior at Critical Points

Now, evaluate the function at points less than, equal to, and greater than \(x=1\). If \(f'(x)>0\), the function is increasing, and if \(f'(x)<0\), the function is decreasing.

Step 4: Conclusion

After analyzing the values of the derivative, we conclude that: the function \(f(x)=x^{2}-2 x\) is decreasing on the interval \(-( \infty,1)\) and increasing on the interval \((1, \infty)\). The behavior of the function changes at \(x=1\).

Key Concepts

Critical Points

Critical points are essential in understanding the behavior of functions. They are where the function's rate of change switches or does not exist. To find them, one takes the derivative of the function and looks for values of the independent variable where this derivative is zero or undefined. For the function \(f(x)=x^2-2x\), we differentiate to obtain \(f'(x) = 2x - 2\). Setting this equal to zero, \(2x - 2 = 0\), we find \(x = 1\) as our critical point. This point is crucial for identifying where the behavior of the function may change, making it a pivotal step in the analysis.

First Derivative Test

The first derivative test is a powerful tool used to determine whether a critical point is a local maximum, minimum, or neither. After finding the critical points, one must examine the sign of the first derivative before and after these points to make conclusions about the nature of the critical point.

If \(f'(x) > 0\) on one side of the critical point and \(f'(x) < 0\) on the other, the point is a local maximum. Conversely, if \(f'(x) < 0\) first and then \(f'(x) > 0\), the critical point is a local minimum. If the derivative has the same sign on both sides, the critical point is neither a maximum nor a minimum. For the given function \(f(x)\), after evaluating the first derivative around \(x = 1\), we observe a transition from negative to positive, indicating an increasing function and thus determining that \(x = 1\) is a local minimum.

Increasing and Decreasing Intervals

To fully comprehend a function's return and its trend, identifying increasing and decreasing intervals is essential. A function is said to be increasing on an interval where its derivative is positive, as the output values rise when moving from left to right. Conversely, the function decreases on intervals where its derivative is negative. This information provides insight into the function's growth and decline.

For the aforementioned function \(f(x) = x^2 - 2x\), the derivative \(f'(x)\) suggests that on the interval \( (-\infty, 1) \) the function is decreasing, as \(f'(x)\) is negative. Beyond the critical point, where \(x > 1\), \(f'(x)\) becomes positive, indicating that the function is increasing on the interval \( (1, \infty) \) and confirming a change in behavior at the critical point \(x = 1\). Thus, understanding the derivative and its sign changes can reveal a great deal about the behavior of the function across different domains.