Find the critical numbers of f (if any). Find the open intervals on which the function is increa

Find the critical numbers of f (if any). Find the open...

Key Concepts

Graphical Verification

Graphical verification involves using tools like graphing utilities or calculators to visually confirm the analytical findings of a function’s behavior. This process helps ensure that the calculated critical points, intervals of increase or decrease, and relative extrema accurately reflect the function’s graph.

Increasing/Decreasing Intervals

Increasing and decreasing intervals refer to portions of the domain where the function's derivative is positive or negative, respectively. Analyzing these intervals is crucial for understanding the overall behavior and trends of the function.

Relative Extrema

Relative extrema are local maximums or minimums of a function, indicating points where the function reaches a peak or a valley within a certain interval. These points are identified by examining the critical numbers and the change in sign of the derivative.

Derivative

The derivative of a function measures its instantaneous rate of change with respect to its independent variable. Calculating the derivative is essential for finding where the function increases, decreases, or achieves relative extrema, forming the backbone of differential calculus.

Critical Numbers

Critical numbers are the values in the domain of a function where its derivative is zero or undefined. They are significant because they indicate where the function could have local maximums, minimums, or other interesting behavior, making them a key part of analyzing the graph of a function.

Quotient Rule

The quotient rule is a method used to differentiate functions that are expressed as the ratio of two differentiable functions. It provides a systematic technique for finding the derivative of rational functions, which is necessary for analyzing their critical points and monotonicity.

Transcript

00:01 For this problem, we are asked to find the critical numbers of the function f of x equals x squared minus 2x plus 1, all divided by x plus 1.

00:08 Then to find the open intervals on which the function is increasing or decreasing, and to locate all relative extrema.

00:15 We are to use a graphing utility to confirm our results.

00:19 To begin, we need to take the derivative of f of x.

00:23 We'll have to apply the quotient rule here.

00:27 So differentiating the numerator gives us 2x minus 2.

00:30 We're multiplying that by x plus 1.

00:33 Then when you have minus the numerator, x squared minus 2x plus 1, times just 1 from the derivative of the denominator, and that all is getting divided by x plus 1 squared.

00:45 So we can end up simplifying the numerator, simply to x squared plus 2x minus 3, divided by x plus 1, all squared.

00:55 Now we can then see that we'll have at least two different critical numbers, two different kinds of critical numbers at least.

01:04 We can see that we'll have the derivative will approach infinity when x equals negative 1, so we need to keep that in mind, as well as the fact that we'll have for the other one here.

01:19 Let's see.

01:20 We'll have that the function will be equal to zero, or not the function, the derivative will be equal to zero, so we'll have a turning point for when x squared plus 2 ,000, x minus 3 equals 0.

01:37 We can find this easily by first factoring the expression, writing it as x minus 1 times x plus 3.

01:46 So we can clearly see that that will be the case when we have x equals 1 and when x equals negative 3.

01:53 Excuse me.

01:54 So plugging in, or if we were to plug in a value that is in between negative 3 and negative 1.

02:01 So let's say f prime of negative 2.

02:04 Then we will find, one moment here, f prime of negative 2 would be negative 3.

02:14 So we can see that we are going to be decreasing in between negative 1 and negative 3...

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