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AP.CALC:

FUN‑4 (EU)

, FUN‑4.A (LO)

, FUN‑4.A.4 (EK)

, FUN‑4.A.5 (EK)

, FUN‑4.A.6 (EK)

so I have the function G here it's expressed as a fourth degree polynomial and I want to think about the intervals over which G is either concave upwards or concave downwards and let's just remind ourselves what these things look like so concave concave upwards is a is an interval you're an interval where you're concave upwards is an interval over which the slope is increasing and it tends to look like an upward-opening you like that and you can see here that the slope over here is negative and then as x increases it becomes less negative it actually approaches zero it becomes zero then it crosses zero becomes slightly positive more positive even more positive so you can see the slope is constantly increasing and if you think about it in terms of derivatives it means that your first derivative is increasing over that interval and in order for your first derivative to be increasing over that interval your second derivative F prime prime of X actually let me write it as G because we're using G in this example in order for your first derivative to be increasing your let me write this so G so concave upward means that your first derivative increasing increasing which means which means that your second derivative is greater than zero and concave downward is the opposite concave downward downward is an interval or you're going to be concave downward over an interval when your slope is decreasing so G prime of X is decreasing or we can say that our second derivative our second derivative is less than zero and once again I could draw it on this so when X is lower we have a look we have a or it looks like we have a positive slope then it becomes less positive and then it becomes less positive approaching zero becomes zero then it becomes negative and then even more negative and then even more negative so as you see our slope is constantly decreasing as x increases here so in order to think about the intervals where G is either concave upward or concave downward what we need to do is let's find the second derivative of G and then let's think about the points at which the second over the second where the second derivative can go from being from going from being positive to negative or negative to positive and that those will be places where it's either undefined or where the second derivative is equal to zero and then let's see what's happening in the interval between and then we'll know what over what intervals are we concave upward or concave downward so let's do that so let's first let's take the first derivative G prime of X it's going to apply the power rule a lot 4 times negative 1 is negative 4x to the 3rd power plus okay you're going to have 2 times 6 is plus 12x to the first power just write it as X and then minus 2 I could say minus 2x to the zeroth power but that's just -2 and then the derivative of negative 3 of a constant is just 0 and now I can take the second derivative G prime prime of X is going to be equal to 3 times negative 4 is negative 12x squared decrement the exponent plus 12 and so let's see where could this be undefined well the second derivative is just a quadratic expression here which would be defined for any X so it's not going to be undefined anywhere so interesting points where we could transition from going from a negative to a positive or positive to a negative second derivative is where this thing could be equal to 0 so let's figure that out so let's figure out where negative 12x plus 12 could be equal to 0 see we could subtract 12 from both sides and we get negative 12x squared is equal to negative 12 divide both sides by negative 12 you get x squared is equal to 1 or take or X could be equal to the plus or minus or X could be equal to the plus or minus square root of 1 which is of course just 1 so at this the second derivative at plus or minus 1 is equal to 0 so either between plus or minus 1 or on either side of them we are going to be we could be concave upward or concave downward so let's let's let's think about this and to think about this I'm going to make a number line let me find it a nice nice soothing color here alright that's a nice soothing color and let's say actually make the number on a little bit bigger so there we go let's utilize the screen space and so if this is 0 this is negative 1 this is negative 2 this is positive 1 this is positive 2 we know that at x equals negative 1 and x equals 1 our second derivative is equal to 0 so let's think about what's happening in between those places to see if our second derivative is positive or negative and from that we'll be able to say where it's concave upward or concave downward so on this first interval right over here so this is the interval from this is the interval where where we're going from negative infinity to negative 1 well let's just try a value in that interval to see whether our second derivative is positive or negative and let's see an easy value there could be negative 2 it's in that interval so let's take G prime prime of negative 2 which is equal to negative 12 times 4 because negative 2 squared is positive 4 so it's negative 48 plus 12 so it's equal to negative 36 the important thing to realize then is well if over here it's negative that over this whole interval because it's not crossing through 0 or it's not discontinuous at any of these points that's why we pick this interval that over this whole interval G prime prime of X is less than 0 which means that over this interval we are concave downwards so concave concave downward concave downward now let's go to the interval between negative 1 and 1 so this is the open interval between negative 1 and 1 and let's try a value there well let's just try 0 will be easy to compute G prime prime of 0 well when X is zero this is zero so it's just going to be equal to 12 the important thing to realize is our second derivative here is greater than zero so we are concave upward concave upward on this interval between negative 1 and 1 and then finally let's look at let's look at the interval where X is greater than 1 so this is the interval from 1 to infinity if we want to view it that way and let's just try to value let's try G prime prime of 2 because that's in the interval and G prime prime of 2 is going to be the same thing as G prime prime of negative 2 because whether you have a negative 2 or positive 2 you squares could becomes 4 so you're going to 4 times negative 12 which is negative 48 plus 12 which is negative 36 which is negative 36 and so once again on this interval you are concave concave downward now let's graph this ahead of time let's see if what we just established is actually consistent with what the graph actually looks like we were able to come up with these these insights about the concavity without graphing it but now it's kind of satisfying to take a look at a graph and actually let me see if I can match up the interval so actually this is pretty closely matched right over here and so this is actually make it a little bit smaller all right and so let me move my bounding box so I'm saying that I'm concave downward between negative infinity negative infinity all the way until all the way until negative 1 all the way until this point right over here so all the way until that point and that looks right it looks like the slope is constantly decreasing all the way until we get to x equals negative 1 and then the slope starts increasing the slope starts increasing from there from there all the way and right at X we're transitioning so I'm going to leave a little I won't color in that and so here our slope is increasing doing that same color our slope is increasing increasing increasing increasing increasing all the way until we get to x equals one and then our slope starts decreasing again and we get back into concave downwards oops I want to do that in that orange color we get back into concave downwards so what we were able to figure out by just taking the derivatives and doing a little algebra we can see quite clearly looking at the graph

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