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# Connecting f and f' graphically

## Video transcript

so what we have plotted here some function let's call it f of X and it's derivative F prime of X what we need to figure out is which one is f of X and which one is f prime of X so let's take a stab at it let's say let's think about what would be the situation if the green function were f of X so let's see if this works out if the green function were f of X does the orange function here the yellow function does that could that be F prime of X so let's think about what's happening to this green function at different points so this green function right over here right at this point if we start at the left has a positive has a positive slope while so if this orange function were f of X it would need to be positive sorry if this orange function where F prime of X if it were the derivative of the green function then it would have to be positive because the green function slope is positive at that point but we see that it's not positive so it's pretty clear that the green function cannot be f of X and the yellow function cannot be it's derivative because if this was its derivative it would be positive here so that quickly we found out we found out that that can't be the case well let's see if it could work out the other way so it's starting to feel just kind of ruling that situation out that maybe that this is f of X and the green function is f prime of X is f prime of X so let's see if this holds up to scrutiny so what we have when we start off at the left f of X what we think is f of X has a reasonably positive slope is that consistent well yeah sure our green function is positive our green function is positive in fact at the point is telling us that the slope of the tangent line is around two and a half and it actually does look like the slope of the tangent line is exactly two and a half of this function right here actually let me erase this so we don't look like we're trying to take the slope of the tangent line of the derivative so it let's just like that so we see the slope of the tangent line right over here looks like about two and a half and the value of this function up here looks like it's about two and a half so so far this green function looks like a pretty good candidate for the derivative for the derivative of this yellow function but let's let's keep going here so let's think about what happens as we move to the right so here let's see it looks like the slope the slope of this yellow function we're just in a color we can see it is it keeps going up it keeps going up keeps going up and then at some point it reaches some maximum slope and then it starts to go down again it starts to go down again the slope starts to go down again all the way to the slope going all the way down to zero right over here well does this green function describe that well see the slope is positive and increasing up to this point which seems pretty consistent with what we just experienced then the slope stays positive but it's positive and decreasing and that's what we saw here the slope is positive and decreasing all the way to the slope getting zero at this maximum point here and we see indeed that on this green function the green function hits zero so it seems like it's doing a pretty good job of plotting the slope of the tangent line of the orange function and then our slope becomes more and more negative our slope becomes more and more negative and then it hits some point some minimum point right over here the slope hits some minimum point right over here and then it becomes less and less negative and then the slope becomes less see how well I can draw it the slope becomes less and less negative until it hits a zero slope again a zero slope again and then it starts becoming positive starts becoming positive until it hits some maximum slope but that stays positive but it becomes less positive it becomes less and less positive so it looks pretty clear that the orange function is f of X and the green function is f prime of X