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Current time:0:00Total duration:4:12

Video transcript

let G be a twice differentiable function defined over the closed interval from negative seven to seven so it includes those endpoints of the interval this is the graph of its second derivative G prime prime so that's the graph right over there y is equal to G prime prime of X and they asked us how many inflection points does the graph of G have so let's just remind ourselves what an inflection point is so that is when we go from being concave downwards to concave upwards so something like this and another way to think about it a point where our slope goes from decreasing to increasing so here our slope is that then it's a little lower but it's a little lower then it's a little lower but then all of a sudden it starts increasing again it starts increasing getting higher higher and higher so that would be an inflection point whatever x value where that would actually happen that would be an inflection point you could go the other way around you could have a function that looks something like this where we have a negative slope but then it's our slope is increasing slope is increasing slope is increasing but then our slope begins decreasing again this two would be an inflection point so in other videos we go into more of the intuition of how do you think about the first and second derivatives of a function at an inflection point but the big picture at least for the purposes of this worked example is to realize when you're looking at the second derivative you have an inflection point where the second derivative crosses the x axis it's not just it's not enough to just touch the x axis you must cross the x axis and so right over here we are crossing the x axis so that is an inflection point right over here we are crossing the x axis so that is an inflection point here and here we touch the x axis our second derivative is equal to zero but we don't cross we don't cross the actual x axis we don't go from being positive to negative we go we stay non-negative this entire time similarly right over here maybe something interesting happens past this point but they're telling us that the function is only defined over this interval so actually nothing happens beyond we getting that points who are not going to cross the x-axis so to answer the question how many inflection points does the graph G have well it has two inflection points looking at the second derivative here now we know the answer why does that make sense why does you have to cross the Y you have to cross the x axis well let's just imagine let's say that this is the graph of a second derivative so this is F prime prime so the first derivative for example could look like this let me the first derivative might look like this where over here we have a negative slope negative slope negative slope negative slope but it's getting closer and closer and then right over here all of a sudden the slope becomes positive and increasing so that would be F prime of X and then you could think about well if this is describing the derivative of our function then what's our function going to look like well our function over here would have a very positive slope but then the slope would keep decreasing all the way up until this point and that it increases again so we have a positive slope right over here so for example our function might look like this it might have a very positive slope but then the slope keeps decreasing and then right over here all of a sudden the slope begins increasing the slope begins increasing again and so here we were concave downward over this first part over this first part we have a positive slope but it's decreasing positive slope but it's decreasing and then we go to having a positive slope but now we are increasing again and so this should give you a good sense for why you need to cross the x-axis in the second derivative