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let F be a differentiable function for all X if F of negative two is equal to three and F prime of X is less than or equal to seven for all X then what is the largest possible value of f of ten and so I encourage you to think about this on your own pause the video try to figure out the largest possible value for f of ten and then we'll work through it together so I'm assuming you've given a go at it so let's let's visualize this so let me draw some axes here so let's say that's my x axis that's my x axis right over there and this right over here is my y-axis that's my y-axis or I'll graph y equals f of X and they tell us F of negative two is equal to three so let's say that this right over here and the two axes aren't going to be drawn to scale so let's say that this is negative two and this right over here is the point negative two comma three and they tell us that F prime of X is less than or equal to seven that the instantaneous slope is always less than or equal to seven so let's let's oh really the the way to get the largest possible value of F we don't have to necessarily invoke the mean value theorem although the mean value theorem will help us know for sure is to say well look the largest possible value of f is if we essentially of F of ten is essentially if we max this thing out if we assume that the instantaneous rate of change just stays at the ceiling right at seven so if we assumed that our function the fastest-growing function here would be a line that has a slope exactly equal to seven so a slope of seven would look and obviously I'm not drawing this to scale this will move visually this looks more like a slope of one but we'll just assume this is slope of seven because it's not at the same this is not they're not the X&Y are not the same scale so slope is equal to seven and so if our slope is equal to seven where do we get to when x is equal to ten when X is to ten which is right over here well what's our change in X so what's our change in X let's just think about it this way our change in Y over change in X is going to is going to be what well our change in Y is going to be f of 10 minus F of 2 f of 2 is 3 so minus 3 over our change in X our change in X is 10 minus negative 2 10 minus negative 2 is going to be equal to 7 this is the way to max out our what our what what our value of f of 10 might be if we did if at any point the slope or anything less than that because remember it can never be the this instantaneous rate of change can never be more than that so if we start off even a little bit lower then the best we can do is get to that we remember we can't we can't do something like that that would get us too steep so it has to be like that we could and then we would get to a lower we would get to a lower f of 10 or if we did if we did some every time you you have a slight a slightly lower rate of change and it kind of limits what happens to you so remember our slope can never be more our slope can never be more than 7 so this part should be parallel so this should be parallel to that right over there this should be parallel but you can never have a higher slope than that so the way to max it out is to actually have a slope of 7 and so what is f of 10 going to be so let's see 10 minus negative 2 that is 12 multiply both sides by 12 you get 84 so f of 10 minus 3 is going to be equal to 84 or F of 10 is going to be equal to 87 so if you have a slope of 7 if you have a slope of 7 the whole way you travel 12 that means you're going to increase by 84 if you started at 3 you increase by 84 you're going to get to 87