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# Inflection points (algebraic)

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.4 (EK)
,
FUN‑4.A.5 (EK)
,
FUN‑4.A.6 (EK)

## Video transcript

let G of X equal 1/4 X to the fourth minus 4x to the third power plus 24x squared for what values of X does the graph of G have an inflection point we'll have a point of inflection so let's just remind ourselves what a point of inflection is a point of inflection is where we go from being cut where we change our concavity or you could say where our second derivative G prime of X switches signs switches switches signs so let's study our second derivative in order to study our second derivative let's find it so we know that G of X is equal to 1/4 X to the fourth minus 4 X to the third power plus 24x squared so given that let's now find G prime of X G prime of X is going to be equal to I'm just going to apply the power rule multiple times 4 times 1/4 is just 1 I'm not going to write the 1 down it's going to be 1 times X to the 4 minus 1 power so 4 to the third power minus 3 times 4 is 12 X to the 3 minus 1 power X to the second power plus 2 times 24 48 X to the 2 minus 1 or X to the first power I could just write that as X so there you have it I have our first derivative now we want to find our second derivative so G prime prime of X is just the derivative of the first derivative with respect to X and so more of the power rule 3x squared minus 24x to the first or just 24x plus 48 so let's think about where this switches signs and this is this is a continuous function it's going to be defined for all X's so the only potential candidates of where it could switch signs are when this thing equals zero so let's see where it equals zero so let's set it equal to zero 3x squared minus 24x plus 48 is equal to zero let's see everything is divisible by 3 so let's divide everything by so you get x squared minus 8x plus 16 plus 16 is equal to zero and let's see can i factor this yeah this would be X minus 4 times X minus 4 or you could just view this as X minus 4 squared is equal to 0 or X minus 4 is equal to 0 so so or where x equals 4 so G prime prime of 4 is equal to 0 so let's see what's happening at on either side of that let's see if G if we're actually if we're actually switching signs or not so let me draw a number line here and so this is so this is 2 3 4 5 and I could keep going and so we know that something interesting is happening right over here G prime prime of 4 is equal to 0 G prime prime of 4 is equal to 0 so let's think about what's what the second derivative is when we are less than 4 and so actually let me just try G prime prime of 0 since that'll be easy to evaluate G prime prime of 0 well it's just going to be equal to 48 so when we are less than 4 our second derivative G prime the second derivative is greater than 0 so we're actually going to be concave upwards over this interval to the left of 4 now let's think about to the right of 4 2 this is a different color so what about to the right of 4 and so let me just evaluate what would be an easy thing to evaluate well I could evaluate G prime of well when I do Jeep or the second derivative is a of 10 so G I'll do it right over here let me do it well I'm running a little bit of space so I'll just scroll down so G prime prime of 10 is going to be equal to 3 times 10 squared so it's 300 minus 24 times 10 so minus 240 plus 48 so let's see this is 60 this is so 300 - 240 is 60 plus 48 so this is equal to 108 so it's still positive so on either side of for G prime prime of X is greater than zero so even this even though the second derivative at x equals four is equal to zero on either side we are concave upwards on either side the second derivative is positive and so and that that was the only potential candidate so there are no X values of X for which G has a point of inflection x equals four would have been a value of x at which G had a point of inflection if we switch if the second derivative switch signs here if it went from positive to negative or negative to positive but it's just staying from positive to positive so the second derivative is positive it just touches zero right here and then it goes positive again so going back to the question for what X values is the graph of G have a point of inflection no X values I'll put an exclamation mark there just for drama
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