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Numerade Educator

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Problem 64 Medium Difficulty

Use a graph to estimate the critical numbers of $ f(x) = | 1 + 5x - x^3 | $ correct to one decimal place.

Answer

Critical numbers where $f'(x)$ does not exist: $-2.1$, $-0.2$, and $2.3$.
Critical numbers where $f'(x) = 0$: $-1.2$ and $1.2$.

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Video Transcript

here we want to estimate graphically the critical numbers of the function F of x equal absolute value of one plus five X minus x. Q, Correent one. The small place. And here we have the graph of the function. Once the inter from negative 626 as we can see, we have uh when we take the absolute value, we have this graph where we uh see this point here for example where there is some kind of big war function is not differentiable. That's the effect of taking the absolute value. We have also here a number where the derivative does not exist. And here, so there are three points where There is an abrupt change and two slopes of the tangents. And therefore these three points I mark here our points for the function is not differential. That is where the derivatives does not exist. And so because they are on the domain of the function they are critical points or critical numbers. We also have Points for the Derivative zero. Around here, around here We have a 10 horizontal tangent line at those wants and it seems that there are no more points for dark or because in fact these phenomenal itself, degree three and we take its absolute value. So these two lines from here to here goes up without any bound. So there won't be another points where this situation we see here, of course. So these are the only five critical numbers of the function and three of them those marked with the red arrows corresponds to points where the relative does not exist and is to hear marked with the blue dots and now with the blue errors Are points where derivative is equal zero. And so they are also critical numbers. So he did some zoom around those points each of them. So we are going to see but more or less the values of the X correspondent to this point. So that is the critical numbers. And here we have one. Let's see, let's use red because I used red for those points. So at this point here it is the first one from left, right where the relative does not exist. That is corresponds to this point here and it's a critical number where the relative that does not exist does not exist uh about eggs, -2.1. Yeah, because we see here the software gave us this decimals and we can see that the decimal One is correct. So we have negative 2.1. So it's our first critical number attitude everyone as we do now for the second critical point. But now in this case for the reality zero, this one I'm uh huh. Yeah, went in here. So we had this one around here As we can see values about native one 0.3 Or we can say Nancy 1.2 because we have 1.3 just around here. So I would say that he derivative zero at about that is critical number where There is a zero at about eggs -1.2 near 91.3 We can say. So now we go for the next one and the next one as we saw in the journal graph is this point here where there is no derivative and we already see this. So I'm going to be. And next one this one here where derivative does not exist and that's about -0.1 very near. In fact it's, We could say it's negative 0.2 because we haven't to write to the sponsor, we have a critical point a number where derivative does not exist at about eggs -0.2. So we had this other critical number. Now the next one will be this Where do the relative zero and let's see this one here. So this is a critical number with to video of F equals zero at about Ex BN 1.3. And yeah uh we have this and now yeah, we go to the last one which is another a point where the derivatives does not exist. This one here. Yeah, we get it This here, this is the last one. So it's critical number where theory of motive of F and X does not exist at about mhm X is 2.3 marvelous. So yeah, summing up all the information we saw in the graphs. So we can say that in summary pretty cool numbers where the first relative does not exist negative 21, -0.2 two 0.3. And critical numbers where The first serve a tive of F0 are negative one point two And 1 3. And this is very close. 2 -1.3, as we said, hear about right here. So these are the five five critical uh numbers in tora or dysfunction.