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# Finding relative extrema (first derivative test)

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.2 (EK)

## Video transcript

in the last video we saw that if a function takes on a minimum a minimum or maximum value min Max value for our function at X equals a then a is a critical point then x equals a is a critical point but then we saw that the other way around isn't necessarily true X X X equal a being a critical point does not necessarily mean that the function takes on a minimum or maximum value at that point so we're going to try to do in this video is try to come up with some criteria especially involving the derivative of the function around x equals a to figure out if it is a cry if it is a minimum or a maximum point so let's look at what we saw in the last video we saw that this point right over here is where the function takes on a maximum value so this critical point in particular was X naught what made it a critical point was that the derivative is 0 you have a critical point where either the derivative is 0 or the derivative is undefined so this is a critical point and let's explore what the derivative is doing as we approach that point so in order for this to be a maximum point the function is increasing as we approach it the function is increasing is another way of saying that the slope is positive the slope is changing but it stays positive the whole time which means that the function is increasing and the slope being positive is another way of saying that the derivative the derivative is greater than 0 as we approach that point now what happens as we pass that point right at that point the slope is 0 but then as we pass that point what has to happen in order for that to be a maximum point well the value of the function has to go down the value of the function is going down that means the slope is negative and that's another way of saying that the derivative the derivative is the derivative is negative so that seems like a pretty good criteria for identifying whether a critical point is a maximum point so let's say that we have critical point critical point a we are at a maximum point we are at a max if if the if f prime of X if f prime of x switches signs switches switches signs from positive to negative from positive to negative as we cross as we cross X as we cross x equals a that's exactly what happened right over here let's make sure it happened at our other maximum point right over here so right over here as we approach that point as we approach that point the function is increasing the function increasing means that the slope is positive it's a different positive slope the slope is changing it's actually getting more and more and more more steep which or more and more and more positive but it is definitely positive so it definitely meets so it's positive going into that point and then it becomes negative as after we cross that point the slope was undefined right at the point but it did switch signs from positive to negative as we crossed that critical point so these both meet our criteria for being a maximum point so so far our criteria seems pretty good now let's make sure that somehow this point right over here which we identified in the last video as a critical point let's make an I think we call this to see this was X 0 this was X 1 this was X 2 this was X 1 this was X 2 so this is X 3 let's make sure that this doesn't somehow meet the criteria because we see visually that this is not a maximum point so as we approach this our slope is negative and then as we cross it our slope is still negative we're still decreasing so we haven't switched signs so this does not meet our criteria which is good now let's come up with a criteria for a let's come up with criteria for a minimum point and I think you could see where this is likely to go well we identified in the last in the last video that this right over here is a minimum point we can see that it's a local minimum just by looking at it and what's the slope doing as we approach it so the function is decreasing the slope is negative as we approach it f prime of X is less than zero as we approach that point and then right after we cross it this wouldn't be a minimum point if the function if the function where to keep decreasing somehow the function needs to increase now so let me do that same green so right after that the function starts increasing again f prime of X is greater than zero so this seems like pretty good criteria for minimum point f prime of x switches switches signs signs from negative to positive as we cross as we cross a if we have critics some critical point a we have a the function takes on a minimum value at a if the derivative of our function switches signs from negative to positive as we cross a from negative to positive now once again this point right over here this critical point X sub three does not meet that criteria we go from negative and to zero right at that point then stay go to negative again so this is not a minimum or a maximum point