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

First Derivative Test

The First Derivative Test is an analytical method used to determine the nature of critical points. By investigating the sign changes of the derivative around a critical number, this test can classify the point as a local maximum, a minimum, or neither, offering insight into the function’s graph and behavior.

Increasing and Decreasing Functions

A function is said to be increasing on an interval if its derivative is positive throughout that interval, meaning the function's output continuously rises. Conversely, it is decreasing if the derivative remains negative, indicating a consistent fall in output. Analyzing the sign of the derivative on different intervals helps to determine these behaviors.

Differentiation

Differentiation is the process of calculating a derivative, which represents the instantaneous rate of change of a function with respect to its variable. This concept is essential in calculus for analyzing and understanding the behavior of functions, especially when determining where functions have peaks, valleys, or change their trend.

Critical Numbers

Critical numbers are values in the domain of a function where its derivative is zero or undefined. These points are crucial because they are potential locations for local maxima or minima and can indicate transitions in the function’s behavior.

Transcript

00:01 Hello, chapter 5, section 1, we're looking at problem 18, which gives us a function f of x equal to 8x cube minus 6x squared minus 72 x plus 2.

00:13 And we're asked to do three things, find the critical numbers, and then describe where the function is increasing, and see where it is decreasing.

00:24 So we start by finding the critical numbers, which occur when f prime of x is equal to zero or undefined.

00:33 Since we have a polynomial here, it's not going to be undefined.

00:37 But let's go ahead and find the derivative first.

00:40 So we're going to use the power rule.

00:42 We're going to multiply by 3, get 24, and subtract one from the exponent.

00:48 So 24x squared, we continue.

00:50 We get minus 12x and minus 7.

00:53 So i'm going to pull out the 12.

00:59 I'm left with 2x squared minus x minus 6 and set that equal to 0 so we can find our critical numbers.

01:11 So keeping my 12, i go ahead and factor.

01:15 So the factors would include 2x plus 3 times excuse me, x minus 2.

01:23 And so if we set each factor equal to 0, we're going to get at x equally negative three halves and x equaling positive two as our critical numbers and that is our answer to part a find the critical numbers so now i want to go ahead and make a sign chart pick the wrong thing there we go so i'm going to make a straight line here and in our sign chart we're going to use our critical numbers, which was negative three halves and positive two.

02:23 So this is coming from f prime is where the critical numbers came from.

02:29 So we're going to look at the signs of f prime.

02:32 So here in factored form is the easiest place to check this out.

02:36 So i can say that if i pick a number to the left of negative three halves, like negative five, 12 is definitely positive.

02:44 And then here we would get negative 10...

Please give Ace some feedback