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# Finding critical points

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

let's say that f of X is equal to X times e to the negative 2x squared and we want to find any critical numbers for F so I encourage you to pause this video and think about can you find any critical numbers of F so I'm assuming you've given a go at it so let's just remind ourselves what a critical number is so we would say C is a critical number critical number of F if and only if if alright if with two F's short for if and only if F prime of C is equal to 0 or F prime of C is undefined is undefined so if we look for the critical numbers for F we want to figure out all the places where the derivative of this with respect to X is either equal to zero or it is undefined so let's think about how we can find the derivative of this so let's see F prime of X F prime of X is going to be well let's see we're going to have to apply some combination of the product rule in the chain rule so it's going to be the derivative of it's going to be the derivative with respect to X of X of X so it's going to be that times e to the negative 2x squared plus plus the derivative with respect to X plus the derivative with respect to X of e to the negative 2x squared of e to the negative 2x squared x times X times X so this is just the product rule right over here derivative of this of the x times e to the negative 2x squared plus the derivative of e to the negative 2x squared times X right over here so what is this going to be well all of this stuff in magenta the derivative of X with respect to X that's just going to be equal one so this first part is going to be equal to e to the negative 2x squared and now the derivative of e to the negative 2x squared over here I'll do this in this pink color so this part right over here that is going to be equal to or just apply the chain rule derivative of e to the negative 2x squared with respect to negative 2x squared well that's just going to be e to the negative 2x squared we're going to multiply that times the derivative of negative 2x squared with respect to X and so that's going to be what negative 4 negative 4x so times negative 4x and of course we have this x over here we have that x over there and let's see can we simplify it can we simplify it at all well obviously both of these terms have an e to the negative 2x squared I'm going to try to figure out where this is either undefined or where this is equal to 0 so let's think about this a little bit so let's see if we factor out an e to the negative 2x squared I'll do that in green we're going to have this is equal to e to the negative 2x squared times we have here 1 1 minus 4x squared 1 1 minus 4x squared so this is the derivative of F now is where would this be undefined or equal to or equal to 0 well let's see e to the negative 2x squared this is going to be defined for any any value of x this part is going to be defined and this part is also going to be defined for any value of x so this is there's no points where this is undefined but let's think about when this is going to be equal to 0 so if this product of these two expressions equaling 0 e to the negative 2x squared that'll never be equal to 0 if you get this exponent to be really I guess you could say very negative number you will approach 0 but you'll never get it to be you'll never get to be zero so this part here can't be zero but if the product of two things are zero at least one of them has to be zero so the only way that we can get F F prime of X to be equal to zero is when 1 minus 4x squared is equal to zero so 1 minus 4x squared is equal to 0 let me rewrite that 1 minus 4x squared is equal to 0 when does that happen and this one we can just solve add 4x squared to both sides you get 1 is equal to 4x squared divide both sides by 4 you get 1/4 is equal to x squared and then what X values is this true right well we just take the plus or minus square root of both sides and you get X is equal to plus or minus 1/2 negative 1/2 squared is 1/4 positive 1/2 squared is 1/4 so it x equals plus or minus 1/2 F prime or the derivative is equal to 0 so let me write it this way f prime of 1/2 is equal to 0 and you can verify that right over here and F prime of negative 1/2 is equal to 0 so if someone asked what are what are the critical numbers here critical critical numbers they are 1/2 and negative 1/2
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