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# Worked example: finding relative extrema

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

## Video transcript

so we have G of X being equal to X to the fourth minus X to the fifth and what we want to do without having to graph G we want to figure out what X values does G have a relative maximum and just to remind us what's going on in a relative maximum so let me draw a hypothetical function right over here so a relative maximum is going to happen so you can visually inspect this is okay that looks like a relative maximum it's kind of top of a mountain or top of a hill these all look like relative maxima and what's in common well the graph the function is going from increasing to decreasing at each of those points is going from increasing to decreasing increasing to decreasing it either of the points or you could say that the first derivative is going from positive to negative so if you look at this interval right over here G prime is greater than zero and then over the next interval when you're decreasing G prime would be less than zero so what we really need to think about is when does when does G prime so let me see relative we care about relative maximum point and so that's essentially asking when does G prime go from positive to negative from from I would for or from G prime greater than zero to G prime less than zero and the values that we could look at are the points are our critical points and critical points are where G prime is either zero or it is undefined so let's think about it where is G prime of X equal to zero G prime of X is equal to zero when let's just take G prime of X you leverage the power rule right here for X to the third power for X to the third minus five X to the fourth minus 5 X to the fourth is equal to zero let's see we can factor out an X to the so we have X to the third times four minus 5x is equal to zero so this is going to happen when X is equal to zero that I couldn't let me not skip steps so this is going to happen when X to the third is equal to zero or 4 minus 5x is equal to zero for X to the third equaling zero well that's only going to happen when X is equal to zero and four minus five X equaling zero we'll add 5x to both sides you get four is equal to 5x divide both sides by 5 you get 4/5 is equal to X so here these are the two places where where our derivative is equal to zero now are there any places where our derivative is undefined well our function right over here is is just a straight-up polynomial our derivative is another polynomial it is defined for all real numbers so these are our two critical our critical points or we could even say critical values now let's think about what G prime is doing on either side of these critical values and I'll draw a little number line here to help us visualize this and so so there we go a little bit of a number line and let's see we care about zero and we care about 4/5 so let's say this is negative 1 this is 0 this is 1 and so we have one critical point at let me do this in magenta we have one critical point here at x equals zero and then we have another critical point I will do this at x equals 4/5 so 4/5 is right around there so that is 4/5 and let's just think about what G is what G prime is doing in these intervals and eat and and these critical points are the only places where we where G prime might switch sides switch signs so let's first think about this let me pick some colors I haven't used yet so let's think about the interval from negative infinity to zero so this is the open interval from negative infinity to zero and we could just plug in a value we could let's try negative 1 negative 1 is pretty straightforward to evaluate so let's see if or you're going to have four times negative one to the third power so that's going to be 4 times negative 1 minus 5 times negative 1 to the fourth power so that's just going to be 1 so let's see this is going to be negative 4 minus 5 which is negative 9 so right over here G prime is equal to negative 9 and so we know over this whole interval since it's to the left of this critical point we know that G prime of X is less than 0 and so our function itself is decreasing over this interval and so we know we need to go from increasing to decreasing so you can already say well we can't go from increasing to decreasing at this critical point because we're already decreasing to the left of it but anyway let's just think about what's happening in the other intervals so in the interval between 0 and 4/5 so that interval right over there so it's between 0 and 4/5 well let's just sample a number there let's say the number I don't know 1/2 might be brilliant forward so we can evaluate G prime of 1/2 G prime of 1/2 is equal to 4 times 1/2 to the third power 1 after the third power is 1/8 so it's 4 eighths or it's just 1/2 minus 5 times 1/2 to the fourth so that's 5/16 - 5/16 and so this is equal to 8 16 minus 5 16 which is equal to 3/16 but the important thing is it's equal to a positive value so in this blue interval right over there and actually let me put 4/5 in a different color so we see that it's not part of that interval so in this in this light blue interval right here between 0 and 4/5 G prime G prime of X is greater than zero so we know our function is increasing and so let's see what's happening to the right of this and the easiest value to try out would just be would just be 1 so let's try out x equals 1 it's in that interval so when x equals 1 I'll just write G prime of 1 is equal to 4 minus 5 4 minus 5 which is equal to negative 1 so G prime of X is less than 0 G prime of X is less than 0 so our function so we could say G is increasing here it is decreasing oh sorry let me be careful G is decreasing here the function itself is decreasing because our derivative is negative then our function is increasing here because our derivative is positive and then our function is decreasing here so at what critical point are we going from from increasing to decreasing what we're doing that at x equals 4/5 so we have a relative maximum at at x equals 4/5 4/5 if they said well where do we have a relative minimum point well that's going to be happen at x equals 0 we're going from decreasing to increasing but we've answered their question of where do we find a relative maximum point
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