For each function, find (a) the critical numbers; (b) the open intervals where the function is i

For each function, find (a) the critical numbers; (b) the...

Key Concepts

Polynomial Analysis and Factoring

For polynomial functions, the process of determining critical numbers and the intervals of monotonicity often involves differentiating the polynomial and then factoring the derivative. Factoring simplifies the task of finding the zeros of the derivative, which is crucial for constructing a sign chart and interpreting the function’s local behavior.

Intervals of Increase and Decrease

Once the critical numbers are established, the sign of the first derivative on the intervals defined by these numbers is analyzed. If the derivative is positive over an interval, the function is increasing there; if it is negative, the function is decreasing. This is a key step in understanding the function’s overall behavior, especially concerning optimization.

Critical Numbers

Critical numbers are values in the domain of a function where its derivative is zero or undefined. These points are significant because they can indicate local maxima, local minima, or saddle points, and they serve as the basis for further analysis of the function's behavior.

First Derivative Analysis

The first derivative of a function measures its instantaneous rate of change, which is used to determine the intervals where the function is increasing or decreasing. By setting the derivative equal to zero and solving for the variable, one finds potential critical points that might correspond to local extrema.

Transcript

00:01 The first thing you want to do is to find the derivative of your function.

00:04 So the derivative of this function is going to be 4x cubed plus 3 times 16.

00:13 That's 48 x squared plus 2 times 64 is 128 x.

00:23 That's your derivative.

00:25 For part a to find your critical points, you need to set your derivative equal to zero and solve for x.

00:32 So i'm going to be using factoring to solve for x in this case.

00:38 I notice that there is a greatest common factor of 4x.

00:42 So i can rewrite this as 4x times x squared plus 12x plus 32 equals 0.

00:51 Now this is just a quadratic function.

00:53 So you could use a quadratic equation quadratic formula to solve or factor it.

00:58 If you know what the factors are and you're comfortable with factory.

01:01 We will end up with 4x times x plus 4 and x plus 8 equals 0, meaning that you have critical points at x equals 0, x equals negative 4, and x equals negative 8.

01:17 So that is part a.

01:21 To get ready for parts b and c, you need to draw a number line and put your critical points on that number line.

01:27 We have 0, negative 4, and 8.

01:31 You need to test some points to determine whether you will get positive.

01:36 Or negative outputs.

01:40 So let's go ahead, pick something to the left of negative 4, like the number negative 5.

01:45 Plug in negative 5 and 2 -year derivative to decide do you get a positive or negative answer.

01:52 And after plugging it in, you should realize you're going to get a negative answer.

01:57 Pick something between 0 and negative 4.

02:00 I'm going to pick negative 1 because that's an easier number to work with...

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