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## Relative (local) extrema

# Worked example: finding relative extrema

AP.CALC:

FUN‑4 (EU)

, FUN‑4.A (LO)

, FUN‑4.A.2 (EK)

## Video transcript

- [Voiceover] So we have g(x) being equal to x to the fourth minus x to the fifth, and what we wanna do
without having to graph g, we want to figure out at 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, okay that looks like a relative maximum. That's kind of a top of a
mountain or top of a hill, these all look like relative
maximum and what's in common? Well the graph, the function is going from increasing to decreasing
at each of those points. It's going from increasing to decreasing. Increasing to decreasing
at 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 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 wrote fror. From g prime greater than zero, to g prime less than zero, and the values that we
could look at or 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 well let's just take g prime of x. We're gonna leverage the
power rule right here, four x to the third power,
four x to the third, minus five x to the fourth. Minus five x to the
fourth is equal to zero. Let's see, we can factor
out an x to the third. So we have x to the third times four minus five x is equal to zero. So this is going to happen
when x is equal to zero. Let me not skip steps. So this is going to happen when x to the third is equal to zero, or four minus five x is equal to zero. For x to the third equaling zero, that's only gonna happen
when x is equal to zero, and four minus five x equaling zero, we'll add five x to both sides, you get four is equal to five x, divide both sides by five,
you get 4/5 is equal to x. So here these are the two places, where our derivative is equal to zero. Now are there any places where
our derivative is undefined? Well our function right over here is just a straight up polynomial. Our derivative is another polynomial, it is defined for all real numbers. So these are our two critical points, or we could even say critical values. Now let's think of 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, little
bit of a number line. Let's see we care about
zero and we care about 4/5. So let's say this is negative 1, this is zero, this is one, 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 prime is doing in these intervals, and these critical points
are the only places 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, let's try negative one,
negative one is pretty straight forward to evaluate. So let's see you have four,
you're gonna have four times negative one to the third power. So that's gonna be four
times negative one, minus five times negative
one to the fourth power. So that's just gonna be one. So let's see this is going to
be negative four minus five. Which is negative nine,
so right over here, g prime is equal to negative nine, 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 zero, 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 at the other intervals. So in the interval between zero and 4/5, so that interval right over there, so it's between zero and 4/5, well let's just sample a number there. Let's say the number, I don't know, 1/2? Might be fairly straightforward. So we can evaluate g prime
of 1/2, g prime of 1/2, is equal to four times
1/2 to the third power. 1/2 to the third power is 1/8. So it's 4/8, or it's just 1/2 minus five times 1/2 to the fourth so
that's 5/16 minus 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 light blue interval right here between zero 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 one. So let's try out x equals one. It's in that interval. So when x equals one, I'll just write g prime of one is equal
to four minus five. Four minus five, which
is equal to negative one. So g prime of x is less than zero, g prime of x is less than zero. 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 cause our derivative is negative. Then our function is increasing here cause our derivative is positive, and then our function is decreasing here. So at what critical
point are we going from increasing to decreasing? Well we're doing that at x equals 4/5. So we have a relative
maximum 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 zero. We're going from decreasing to increasing, but we've answered their question of where do we find a
relative maximum point.