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we're asked at which points on the graph is f of x times f prime of x equal to zero so if I have the product of two things and it's equal to zero that tells us that at least one of these two things need to be equal to zero so first of all let's see are there any points when f of X when f of X is equal to zero so we're plotting f of X on the vertical axis we could call this graph right over here we could say this is y is equal to f of X so at any point does the Y value of this curve equal zero so it's positive positive positive positive positive positive but it is decreasing right over here what's decreasing here then it's increasing then it's decreasing and it does get to zero right over here but that's not one of the labeled points and they want us to pick one of the label points or maybe even more than one of these label points so we're going to focus on where an f prime of X is equal to zero and we just have to remind ourselves what F prime of X even represents F prime of X represents the slope of the tangent line at that value of x so for example F prime of 0 which is the x value for this point right over here it's going to be some negative value it's the slope of the tangent line similar similarly f prime of X when X is equal to 4 that's what's going on right over here that's going to be the slope of the tangent line it's going to be a positive value so if you look at all of these where is the slope of the tangent line 0 and what does a zero slope look like well it looks like a horizontal line so where is the slope of the tangent line here horizontal well the only one that jumps out at me is point B right over here it looks like the slope of the tangent line would indeed be horizontal right right over here or another way you could think of it is the instantaneous rate of change right at right of the function right at x equals two looks like it's pretty pretty close to if this is x equals two looks like it's pretty close to zero so out of all the choices here I would say only B looks like the derivative at x equals 2 or 1 or the derivative or the slope of the tangent line at B it looks like it's 0 so I'll say B right over here and then they have this kind of crazy wacky expression here f of X minus 6 over X what is that greatest in value and we have to interpret this we have to think about what is f of X minus 6 over X actually mean whenever I see expressions like this especially if I'm taking a differential calculus class I would say well this looks kind of like finding the slope of a secant line whenever in fact all of the all of what we know about derivatives is finding the the limiting value of the slope of a secant line this looks kind of like that especially if at some point my Y value is a 6 here then this could be the change in Y value and if the corresponding x value is 0 then this would be f of X minus 6 over X minus 0 so do I have 0 6 on this curve here well sure when X is equal to 0 when X is equal to 0 we see that f of X is equal to 6 so what this is right over here let me rewrite this this we could rewrite as f of X f of X minus 6 f of X minus 6 over over X minus 0 X minus 0 so what is this what does this represent well this is equal to the slope this is equal to do submit color this is equal to the slope of the secant line secant line between between the points between the points X f of X X and whatever the corresponding f of X is and and we could write it as 0 f of 0 because we see f of 0 is equal to 6 this right over here is f of 0 in fact let me just write that as 6 and the point 0 6 and the point 0 6 so let's go through each of these points and think about the slope of the secant line between those points are and point a this is essentially the slope of secant line between some point X f of X and essentially Point a so let's draw this out so between a and B you have a fairly negative you have a fairly negative slope remember we want to find the largest slope so here it's fairly negative between a and C it's less negative a and C it's less negative between a and D it's even less negative it's still negative but it's less negative it's less negative and then between a and E it becomes more negative now it becomes more negative and then between a and f it becomes even more negative it becomes even more negative between a and f so when is the slope of the secant line between one of these points and a the greatest or I guess we could say the least negatives because they're always there it seems like they're always negative it would be between point D and a so when is this when is this greatest in value well when we're looking at Point Point D at Point D X is equal to 6 X is equal to 6 it looks like f of X is like five and a half or something so this will turn into f of f of 6 which is five and a half or maybe it's even less than that five and a third or something minus six over six minus zero that's how we'll maximize this value this is the least negative slope of the secant line