WEBVTT
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So in exercise 14 we worked with function F of
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X is equal to two x plus three squared times
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next, minus two to the fifth, All over
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X cube time X minus five squared. And what
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we want to do is to find the first and
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second derivatives, preferably using some kind of cast software
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and then use their grafts to estimate intervals where the
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function is increasing, increasing and buying contact. Body
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of All right. So, um, when we
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actually worked out exercise 14 you just using the graph
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of Ethel X two. Get approximations. We know
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that we should have minimums at X is equal to
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negative three. House and X is equal to about
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eight. So we're just gonna keep that in my
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just to make sure we get the same values for
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this. No, we should have a vertical ass
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into it right here at X is equal to zero
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as well. Over here, X is equal to
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five. So she's going to put those in.
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And I just went ahead and already found a crime
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using a derivative calculator because doing that by hand would
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be way too much work for any sane person.
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And then I went ahead and graft it, and
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I made sure I included all my ex intercepts and
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all of that. So with this here, you
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will see that. So this here was around Negative
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three hats. This here is that, too.
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And then over here, that value was about eight
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. Now, to figure out where the function is
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increasing, we want to determine that F prime of
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X is strictly larger than zero. So this here
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will go off to date, if any. If
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you were to look at the rest of the graph
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to the left of negative three house so to the
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ray is gonna be negative. 3/2 to our first
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vertical awesome being zero in union and then from zero
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to four or too excessive. 505 are next intercept
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. It will be increasing. And then after this
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, we will deposit from eight to and and maybe
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I should just roll that over here. The behavior
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of it looks like that. So it's decreasing or
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it's negative to the left of eight and then pause
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it to the right. All right, so we
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have our interval of increasing. And now to find
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where it's decreasing and let me go ahead and write
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that about here. This is increasing now where the
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function is decreasing. That'll be where prime of X
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is strictly less than zero, and that would just
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be the rest of our interval. So from negative
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22 negative three hops union and then from 5 to
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8, it'll be decreasing. And so let's just
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go ahead and check that the derivative here actually does
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give us two minimums at exited with a negative three
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house and about eight so into negative three house.
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The function should be decreasing and then increasing. Doctor
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. So that is a minimum at X is equal
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to well, it's increasing into the point and then
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increasing after. So that's gonna be a saddle.
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So we don't have any issues there, and then
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at X is equal to about eight. Well,
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it's decreasing in the point and an increasing after.
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So at least our graph here matches up with what
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we would expect for the maxes and meds. All
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right, then we want to find Kong Cavity.
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So began for the second derivative. I just went
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ahead and plugged it into a derivative calculator, and
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it gave me that. Ah, big, monstrous
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looking thing. And then I just kind of zoomed
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out far enough to kind of get a good idea
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of what our behavior should look like. And this
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is what I ended up with for my graph.
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So for us to find Kong cavity, remember,
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we want to figure out where the function he's going
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to be larger than zero for it to be Kong
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cape up. So f double time of strictly larger
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than zero. Uh, and again, I went
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ahead and put in the vertical Assam coats. Just
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we kind of keep track of them. So it
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looks like from 0 to 5, a double prime
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not to bury. You're five. It looks like
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it is positive. And then on the other side
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of our asking toe, it will be positive.
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That's what. So it's concave up from 0 to
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55 30 then forward to be Kong cave down.
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We want to find where the second derivative is less
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than zero, and that would just end up being
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from negative infinity to zero. And if you were
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to go ahead and actually use this information or at
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least used to look at the work we get in
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number 14 you'll see that it pretty much matches up
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pretty well with what we have here.