Find all critical points of the function fx 5x 4x 1 use...

Name: find all critical points of the function fx 5x 4x 1 use symbolic notation and fractions where needed give your answer in the form of a comma separated list if the function does not have any 60382
Uploaded: 2022-05-31T21:37:13-08:00
Duration: 6 min 21 s
Channel: Victoria Chayes
Description: find all critical points of the function fx 5x 4x 1 use symbolic notation and fractions where needed give your answer in the form of a comma separated list if the function does not have any 60382

Find all critical points of the function $f(x) = 5x + |4x + 1|$. (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.) critical points: Question Source: Rogawski's Calculus Early Transcendentals | Publisher: W.H. Freeman

Transcript

00:01 All right, so we want to consider the function f of x equals absolute value of x over x squared plus one, and we're going to be looking at various critical points.

00:10 So now the first thing i like to do when i see a function like this is first of all, we can sketch it.

00:18 So x over x squared plus one, right? when it's positive, it's going to start at zero here.

00:25 It kind of peaks up to one here and then goes down.

00:31 And let's actually, let's make this a little bit more clear.

00:35 It is going down to zero, but it's kind of going down to zero at a slower rate.

00:39 And then because it's absolute value and the denominator is positive, it's going to be doing the same thing here.

00:48 So we are going to be asked to analyze this function for its critical points.

00:53 So first of all, i think it's, again, a little.

00:55 Bit easier to write this as piecewise as x over x squared plus one.

01:02 But now we want this to be negative to get the absolute value positive, right? when x is less than zero, zero, when x is equal to zero, and x over x squared plus one when x is great over zero.

01:17 Okay.

01:19 So now we want the overall critical points so that we can look at extreme values and so the critical points are when f prime of x equals zero or does not exist so immediately we have one at x equals zero right because this absolute value makes a discontinuity in the derivative but then secondly we want to take f prime of x separately for x less than zero and x greater than 0.

01:52 So if we look at d over d x of negative x over x squared plus 1, right? you get, so for the negative side, you will get an x squared, the negative one, yeah, you'll get an x squared minus 1 over x squared plus 1 squared.

02:19 This is just the product rule.

02:21 And the positive one of x over x squared plus 1, you are going to get 1 minus x squared over x squared plus 1 squared.

02:35 This is x greater than 0.

02:37 And so here you can see the denominators are never zero.

02:44 This thing begins when x is negative.

02:48 It begins increasing the moment that x is less than negative one...