Matching functions & their derivatives graphically

I have a function f of X here and I want to think about which of these curves could represent F prime of X could represent the derivative of f of X well to think about that we just have to think about well what is the slope of the tangent line doing at each point of f of X and see if this corresponds to that slope if the value of these functions correspond to that slope so we can see when X is equal to negative 4 the slope of the tangent line is essentially vertical so you could say it's not really defined there but as we go slightly to the right of x equals negative 4 we just have a very very very positive slope so you could kind of view it as our slope is going from infinity to very very positive 2 a little bit less positive 2 a little bit less positive 2 a little bit less positive 2 a little bit less positive so which of these graphs here have that property remember this is trying to graph the slope so which of these which of these functions down here which of these graphs should have a value that is essentially kind of approaching infinity as when X is equal to negative negative 4 and then it gets less and less and less positive as X goes to 0 so this one it looks like it's coming from negative infinity it's getting less and less and less negative so that doesn't seem to meet our constraints this one looks like it is coming from a positive infinity and it's getting less and less and less positive so that seems to be okay this has the same property it's getting less and less and less positive this one right over here starts very negative and gets less and less and less negative so we can rule that out now let's think about what happens let's think about what happens when X gets to 0 when X gets to zero the tangent line is horizontal we're at a maximum point of this curve right over here a slope the slope of a horizontal line is 0 and so which one of these remember we're trying to look for which one of these curves represent the value of that slope so which one of these curves hit zero when X is equal to 0 well this one doesn't so the only candidate that we have left is this one and this one does hit 0 when x equals 0 and let's see if it keeps satisfying what we need for f prime of X so after that point it should start getting more and more negative the slope should get more and more and more negative essentially approaching negative infinity negative infinity as X approaches 4 and we see that here the this value of this function is getting more and more negative and it's approaching negative infinity as X approaches 4 so we'll go with this one this looks like a pretty good candidate for F prime of X