Problem 37 - Essential Calculus

Question

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=\frac{x^{2}-2 x+1}{x+1} $$

Step-by-Step Solution

Answer

The function \(f(x) =\frac{x^{2}-2x+1}{x+1}\) has one critical number at \(x = -1\). The function is increasing on the intervals \((-\infty, -1)\) and \((-1, +\infty)\). The function has no relative extrema. This is confirmed by graphing the function.

Step 1: Simplify the Function

The function \(f(x) =\frac{x^{2}-2x+1}{x+1}\) can be simplified to \(f(x) = x - 1\). This is achieved by dividing both terms in the numerator by \(x+1\), which yields \(\frac{x^{2}}{x+1} -\frac{2x}{x+1} +\frac{1}{x+1}\). The first two terms simplify to \(x-1\), while the third term goes to zero as \(x\) approaches infinity.

Step 2: Find the Derivative

The derivative of the simplified function can be easily calculated, \(f'(x) = 1\).

Step 3: Find the Critical Numbers

Critical numbers are the values of \(x\) that make the derivative zero or undefined. For this function, \(f'(x) = 1\) everywhere except at \(x = -1\), which makes the denominator of the original function zero. Therefore, \(x = -1\) is the only critical number.

Step 4: Determine Intervals of Increase and Decrease

Because \(f'(x) = 1\) for all \(x\neq-1\), the function is increasing on the intervals \((-\infty, -1)\) and \((-1, +\infty)\). There are no intervals on which the function decreases.

Step 5: Locate Relative Extrema

The relative extrema of a function occur at critical numbers where the function changes from increasing to decreasing, or vice versa. In this case, there is only one critical number, \(x = -1\), but the function does not change from increasing to decreasing, or vice versa, at this point. Therefore, the function has no relative extrema.

Step 6: Graph the Function

Use a graphing utility to plot the function \(f(x) = \frac{x^{2}-2x+1}{x+1}\). You should see a graph of a straight line \(x -1\), except at \(x = -1\), which is a hole.

Key Concepts

Increasing and Decreasing Intervals

Understanding the increasing and decreasing behavior of functions is essential in calculus. A function is said to be increasing on an interval if the function values get larger as the input values move from left to right within that interval. Conversely, a function is decreasing when the output values become smaller over the range of input values in the interval. To determine these intervals for a function like ...
When analyzing our function, ...

Relative Extrema

Relative extrema, often called local maxima and minima, are points where a function reaches a peak or valley within a certain interval. A relative maximum is found where the function's value is higher than all surrounding values, and a relative minimum is at a point where it is lower. Identifying these points is a key step in understanding the overall shape and behavior of a function. The given function ...
Relative extrema can often be found by locating the ...

Derivative of a Function

The derivative of a function represents the rate of change or the slope of the function at any point. It is a fundamental tool in finding critical numbers, as well as determining increasing and decreasing intervals and extrema of functions. The derivative is calculated using limits or rules of differentiation, and a function may have a derivative that is constant, variable, positive, negative, zero, or undefined at different points. For the function ...
In our example, the simplified form of the function makes it easy to see that the derivative is constant, which duly affects the number of critical numbers and the increasing/decreasing nature of the function over its domain.