7. DETAILS PREVIOUS ANSWERS 1/2 Submissions Used Determine...

7. DETAILS PREVIOUS ANSWERS 1/2 Submissions Used Determine whether Rolle's theorem can be applied to $f$ on the closed interval $[a, b]$. (Select all that apply.) $f(x) = (x - 3)(x - 6)(x - 7)$, $[3, 7]$ Yes, Rolle's theorem can be applied. No, because $f$ is not continuous on the closed interval $[a, b]$. No, because $f$ is not differentiable on the open interval $(a, b)$. No, because $f(a) \neq f(b)$. If Rolle's theorem can be applied, find all values of $c$ in the open interval $(a, b)$ such that $f'(c) = 0$. (Enter your answers as a comma-separated list. If Rolle's theorem cannot be applied, enter NA.) c = NA Submit Answer

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00:01 Okay so for this question i have put up a graph of the equation in question for your benefit because it's going to make it a little bit easier to answer some of these questions.

00:13 So is f continuous on the closed interval 4, 7? yes absolutely it is continuous, there's nothing that we need to worry about here, there's no kind of funny operations, there's no 1 over x, it's not piecewise, everything is just behaving very nicely, so yes it's continuous.

00:31 Is it differentiable? yeah absolutely it's differentiable, there's no reason for it not to be and if you want to differentiate it there's a couple of different ways you could do this.

00:41 You could expand out the brackets and then differentiate it using the power rule or you can differentiate it using the product rule and that's the way i'll do it.

00:51 So i'm going to differentiate the first bracket x minus 4 and that just becomes 1 and so then we need to multiply that by x minus 6, x minus 7 and then i'm going to differentiate the second bracket which is just 1 and so we need to multiply that by x minus 4, x minus 7 and then i differentiate the third bracket and once again that gives me 1 and so i multiply that by x minus 4, x minus 6 and now you can leave it like this or you can expand out the brackets and collect the like terms.

01:25 So if we expand out the brackets we get x squared and then we have a minus 7x minus 6x that gives me minus 13x and then we have a minus 6 times a minus 7 so that gives me a plus 42 and then we have a plus x squared minus 7x minus 4x gives me minus 11x and then plus 7 times 4 gives me plus 28 and then we have a plus x squared minus 6x minus 4x gives me minus 10x and then plus 24.

02:02 Okay and if we collect all these terms together we'll get 3x squared and then we need to do minus 13x minus 11x minus 10x so minus 13x minus 11x is minus 24x minus 10x is minus 34x and then plus 42 plus 28 plus 24 and that should give you a plus 94 and that's your derivative.

02:34 Okay and so now we want to find f of 4 and f of 7.

02:39 Now you should be able to see from the graph what those values are going to be but if we find f of 4 all we need to do is put 4 into the original equation at the top and so if you put 4 in you're going to get 4 minus 4, 4 minus 6, 4 minus 7.

02:59 It doesn't really matter what the second two brackets are because the first bracket is 0 and so that's going to make the whole thing 0 and we can see that that's that point right there and likewise you can use the same argument for f of 7.

03:11 You're going to end up with a bracket that's so f of 7 equals 0...

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