If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Critical points introduction

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.C (LO)
,
FUN‑1.C.1 (EK)
,
FUN‑1.C.2 (EK)
,
FUN‑1.C.3 (EK)

## Video transcript

I've drawn a crazy-looking function here in yellow and what I want to think about is when this function takes on maximum values and minimum values and for the sake of this video we can assume that the graph of this function just keeps getting lower and lower lower as X becomes more and more negative and lower and lower and lower as X goes beyond the interval that I have depicted right over here so what is the maximum value that this function takes on well we can eyeball that it looks like it's at this point it looks like it's at that point right over there so we would call this a global a global maximum a global maximum the function never takes on a value larger than this so we could say that we have a global maximum at the point at the point X naught because f of f of X naught f of X naught is greater than or equal to f of X for any other X in the domain and that's pretty obvious when you look at it like this now do we have a global minimum point the way that I've drawn it well no this function can take on arbitrarily arbitrarily negative values again it approaches negative infinity as X approaches negative infinity it approaches negative infinity as X approaches positive infinity so we have let me write this down we have no global we have no global minimum now let me ask you a question do we have local minima or local maximum when I say minima is just the plural of minimum and maximum is just the plural of maximum so we have a local minima here or local minimum here well a local minimum you could imagine means that that value of the function at that point is lower than the points around it so right over here it looks like we have a local minimum local we have a local minimum and I'm not giving you a very rigorous definition here but one way to think about it is we can say that we have a local minimum point at x 1.is if we have a region around x1 where f of x1 is less than an f of X for any X in this region right over here and it's pretty easy to eyeball to this is a low point for any of the values of f around it right over there now do we have any other local minima well it doesn't look like we do now what about local Maxima well this one right over here let me do it in let me do it in purple I don't want to get people confuse to actually let me do it in this color this point right over here looks like a local maximum local not lakhs that would have to deal with salmon local local maximum right over there local maximum so we could say at the point X 1 at the point I'm sorry at the point X 2 we or we have a local maximum point at X 2 because f of X 2 is larger than f of X for any X around a neighborhood around X 2 I'm not being very rigorous but you can see it just by looking at it so that's fair enough we've identified all of the maxima and minima often called the extrema for this function now how can we identify those if we knew something about the derivative of the function well let's look at the derivative at each of these at each of these points so at this first point right over here if I were to try to visualize the tangent line let me do that in a better color than brown if I were to try to visualize the tangent line it would look something like that so the slope here is 0 so we would say that f prime of x naught is equal to 0 the slope of the tangent line at this point is 0 what about over here well once again the tangent line the tangent line would look something like that so once again we would say F Prime at X 1 at X 1 is equal to 0 what about over here well here the tangent line is actually not well-defined we have a positive slope going into it and then it immediately jumps to being a negative slope so over here F prime of X 2 is not defined not let me just write undefined undefined so we have an interesting and once again I'm not rigorously proving it to you I just want you to get the intuition here we see that if we have some type of a dreama and we're not talking about when we're when X is at an endpoint of an interval just to be clear what I'm talking about when I'm talking about X as an endpoint of an interval we're saying let's say that the function is let's say where you have an interval from there so let's say a function starts right over there and then keeps going this would be a maximum point but it would be an endpoint we're not talking about endpoints right now we're talking about when we have points in between or when we when when our interval is infinite so we're not talking about we're not talking about points like that or points like this we're talking about the points in between the points in between so if you have a point inside of an interval it's going to be a minimum or maximum and we see the intuition here if you have sewn on endpoint non endpoint non endpoint min or max min or max at let's say X is equal to a so if you know that you have a minimum or maximum point at some point X is equal to a and X isn't the endpoint X isn't the endpoint of some interval this tells you something interesting or at least we have the intuition we see that the derivative at X is equal to a is going to be equal to 0 or the derivative at X is equal to a is going to be n is going to be undefined and we see that in each of these cases derivative 0 derivative 0 derivative is undefined and we have a word for these for these points where the derivative is either 0 or the derivative is undefined we call them critical points we call them critical critical points so for the sake of this function the critical points the critical point the critical points are if we could include X sub 0 we could include X sub 1 at X sub 0 and X sub 1 the derivative is 0 and X sub 2 where the function is undefined now so if we have a non endpoint minimum or maximum point then it's going to be a it's going to be a critical point but can we say it the other way around if we find a critical point where the derivative is 0 or the derivative is undefined is that going to be a maximum or minimum point and to think about that let's imagine let's imagine this point right over here so let's call this X sub 3 if we look at the tangent line right over here if we look at the slope right over here it looks like f prime of X sub 3 is equal to 0 so based on our definition of critical point X sub 3 would also be a critical point but it does not appear to be a minimum or a maximum point so a minimum of a maximum point that's not a tan that's not an end point it's definitely going to be a critical point but a critical point but being a critical point by itself does not mean you're at a minimum or maximum point so just to be clear at all of these points were at a minimum or maximum point this we're at a critical point all of these are critical points but this is not a minimum or maximum point and the next video will start to think about how can you can how you can differentiate or how you can tell whether you have a minimum or maximum at a critical point
AP® is a registered trademark of the College Board, which has not reviewed this resource.