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We're going to talk about the concavity of a graph. The concavity has to do with which
direction the graph curves. If your graph looks like any part of a smiley face, we call
it concave up. If it looks like any part of a frowney face, we call it concave down. Another
way to think of it is, if it would hold water, it's concave up. If it looks like an umbrella,
it's concave down. Mathematically we can define this a couple ways. We can say a function
is concave up if it lies above all of its tangent lines, or we can say it's concave
up if its first derivative is increasing. If the slopes of the tangent lines keep getting
bigger as we go. It turns out these are the same. The same thing for concave down. It's
concave down if it lies below all of its tangent lines or if the tangent lines slopes keep
getting smaller. For example, can we figure out where this graph is concave up and where
its concave down? From little A to little B, it's like a frowney face, so this is concave
down. Then from A to B, it would hold water, it's like a smiley face. It's concave up.
From C to D it's bending back down again. Concave down. D to E, concave up. You might
think you see a pattern there, down, up, down, up. Here we're going to trick you. From E
to P, it's also concave up. It's like part of a smiley face. Then from P to Q, it's part
of a frowney face, so it's concave down. What does concavity have to do with the second
derivative? Well, it turns out the second derivative is a really good way to determine
the concavity of a function. That's because, remember we said that it's concave up if the
slopes of the tangent lines are increasing. We said if F prime is increasing. What does
that mean? That means F prime's derivative, which happens to be F double prime, would
be positive because if the derivate is positive, then the function is increasing. F prime's
derivative which is F double prime, whenever that is positive, F prime's increasing, and
our original function is concave upward. Similarly, if the slopes of the tangent lines, F prime
is decreasing, then F double prime, the derivative of the first derivative is negative, and our
original function is concave down. The test for concavity, wherever the second derivative
is positive, we have concave up. Wherever the second derivative is negative, we have
concave down. Let's try it on this example. First thing we need to do is find the second
derivative. F prime is, we need to use the chain rule here, E to the 2 X, times the derivative
of 2 X which is 2, minus the derivative of E to the X which is itself. Then we do it
again. F double prime of X, this 2 is a constant so it stays, the derivative of E to the 2
X is E to the 2 X times 2 again, minus the derivative of E to the X, which is just E
to the X. Now we need to know when the second derivative changes signs. Well, when is that?
It can only change signs, because it's a nice, pretty, continuous I'm assuming function,
if it is equal to 0 or it's undefined. It's never undefined, so we need to figure out
when it's equal to zero. What do we have here? We have 4 E to the 2 X, minus E to the X equals
0. This maybe kind of looking strange because of these E to the X's, but remember E to the
2 X is the same thing as E to the X squared. We can just factor out an E to the X just
like we would if this was X. We get an E to the X pulled out front, and we're left with
4 E to the X minus 1 equals 0. That means the points that we care about are when E to
the X equals 0, or when 4 E to the X minus 1 equals 0. Can you ever raise a positive
number to a power and get 0? No you can't do that, so that one doesn't make any sense.
Now we have 4 E to the X equals 1. E to the X equals one-fourth. Taking the natural log
of both sides, we get X equals the natural log of one-fourth. Now we're going to make
an F double prime number line. We're going to put on it the natural log of one-fourth,
and we need to plug in numbers on either side of the natural log of one-fourth. It turns
out the natural log of one-fourth as a decimal is approximately, negative 1.38. We can plug
into the first interval, F double prime negative 2, and into the second interval, F double
prime of 0. Let's see if we can figure those out. We have F double prime of negative 2
is going to be E to the negative 2 time 4 E to the negative 2, minus 1. We know E to
the negative 2 is a positive number, because E to anything is a positive number. What about
4 E to the negative 2, minus 1? Let's get out our calculator. That's a negative number.
It's about negative .46 approximately. A positive times a negative here, gives us a negative.
If we do F double prime of 0, that gives us E to the 0, times 4 E to the 0 minus 1. That
one is easy to calculate. E to the 0 is 1, 4 times E to the 0 is just 4, minus 1 is 3,
so this is 3, which is definitely a positive number. That tells us that our original function
F is concave up, from natural log of one-fourth to infinity, and F is concave down from negative
infinity to natural log of one-fourth. Now we can define an in inflection point to be
a point on the graph like this one, where the concavity changes. This is a point that's
on the original curve F of X here, it has to be in the domain of F of X, and you must
have you're concavity changing from negative to positive, or positive to negative. In the
example we just did, natural log of one-fourth is an inflection point. Now we can also find
out things about concavity by looking at the F prime graph. Let's go back to the same graph
we were looking at in the previous video, and let's see if we can figure out where F
is concave up and concave down, based on this F prime graph. If we want F concave up, we
want F double prime to be positive, which means F double prime is the derivative of
F prime. So if something has a positive derivative, then it should be increasing; F prime should
be increasing. If we look at our graph, where is F prime increasing? It's increasing from
negative infinity up to 3 and a half, from negative 2 to 0, and then from 4 to infinity.
F is concave up from negative infinity to 3 and a half, and from negative 2 to 0, and
from 4 to infinity. Similarly, if we're looking for F concave down, then that is where F double
prime is negative. F double prime is the derivative of F prime, so if the functions derivative
is negative, then that function is decreasing. F prime is decreasing, and that's going to
happen from negative 3 halves, to negative 2, and then from 0 to 4. Negative 3 and a
half to negative 2, and from 0 to 4. Now it also asks us to find the inflection points
of F. Remember the inflection points are when the second derivative changes sign. When the
second derivative changes sign, we have a second derivative number line, we can see
they're going from positive to negative or from negative to positive; this is, remember,
the first derivative number line for F prime. This is exactly when F prime has a local max
or a local min. On our graph, this is going to be at negative 3 halves, negative 2, 0,
and 4. Now remember these are inflection points for F. Which means they are local maxs and
mins for F prime, and they are where F double prime goes from positive Y values to negative
Y values, or negative Y values to positive Y values.