Consider the function f x x312x227x 5 show your work a find...

Transcript

00:01 Suppose we have f of x, which is equal to x cubed minus 12x squared minus 27x plus 5.

00:08 The first part of this problem is defined where the function is increasing or decreasing.

00:14 So the first thing you have to do is to find the first and the second derivative of this function.

00:21 Now f -prem of x, this is just equal to 3x squared minus 24x minus 27.

00:30 And the second derivative f double prime of x that's equal to 6x minus 24.

00:37 Now we want to find the critical points of this function.

00:44 And to do this, we have to set the first derivative to zero and solve for x.

00:50 Now, if f prime of x equals 0, we get 3x squared minus 24x minus 27, that's equal to 0.

00:59 We want to factor out 3.

01:01 We get x squared minus 8x minus 9.

01:05 That's equal to 0.

01:06 We get 3 times x minus 9 times x plus 1 equal to 0, meaning x equals 9 or x is negative 1.

01:18 So for the first part, to get the intervals where the function is increasing or decreasing, we will partition our domain negative infinity to infinity using the critical points.

01:35 So if we have this number line, and this is from negative infinity to infinity, we'll partition this using the critical points, negative one, and then let's say nine is here.

01:48 So then we can form the intervals.

01:51 That will be from negative infinity to negative one.

01:55 From negative 1 to 9.

01:58 And then lastly we have 9 to infinity.

02:02 And then we have to pick values inside the intervals.

02:07 So let's say we want to pick negative to here.

02:10 Over here we can pick 0 and over here we can pick 10.

02:14 And then we will find the sign of f prime over each interval.

02:22 And then after that we will use first derivative test.

02:25 So let's look at the sign of f prime when x is negative 2.

02:31 We can use the factored form of f prime.

02:35 So we can use this to get the sign.