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in order to calculate points of
curvature so we're talking about maximum
minimum or points of inflection we now
need to use the second derivative now
we've done the second derivative a few
times through this course but without
really talking about why we're doing it
and this is really important because
this tells us what happens to the curve
after the stationary point so whether
after the stationary point the curve
goes up or whether after the stationary
point the curve goes down or whether it
continues with the same gradient it's
the second derivative that allows us to
find that we often say it measures how
concave a curve is so let me just show
you what I mean in a minimum point you
would have a point like this so let's
just call this our stationary point and
after the point it goes up in order for
this to be a minimum this means that the
second derivative here d2y DX squared
has got to be positive because remember
this is a positive gradient so anything
that goes upward is positive anything
that goes down is a negative gradient so
in this case D 2 y DX squared is greater
than or equal to 0 because it's positive
and this kind of curve if I draw it
properly gives us a minimum curve as
we've just discussed on the previous
slide similarly if you have a stationary
point and straight after the stationary
point the curve goes downwards we see a
negative gradient we can say here that
the second differential D 2 y DX squared
must now be less than zero we call this
point a maximum point as we've just
discussed so these are just little
visual features that we'll have to have
familiarize ourselves with with a point
of inflection we have already said that
we have a flat point so it could be
positive and positive gradients remember
we're talking about and it could be flat
and negative and negative that should be
a little bit more curvy because we're
talking about curves so this here is a
point of inflection
now let's just discuss how we do this
numerically you've found the gradient by
doing dy by DX of a function u equal it
to zero because remember that at
stationary points so I'm just going to
shorten this at stationary points dy by
DX equals to zero as we've stated
earlier so you've found your stationary
points or the coordinates of the
stationary points by equaling it to zero
you then want to know what kind of
stationary point ax is all you do you
take your dy by DX and you differentiate
it the second time that gives you your
second differential here once you know
your second differential you substitute
your X or Y values in there or both and
then you look at whether you get a
positive answer a negative answer or a
zero okay so in this point here in our
points of inflection these are the only
points where we say but d2y DX squared
equal to zero and then this requires a
little bit more investigation and we
investigate by using gradients before
and after the point so let me just
highlight these for you here we are
saying that if we have a second
derivative greater than zero you will
have a minimum point if the second
derivative is less than 0 you will have
a maximum point and if the second
derivative is equal to zero we have a
point of inflection which we will
investigate a little bit further and
I'll show you how that's done in an
example so just to summarize stationary
points can be found by doing dy by DX
equals to zero you will see that we'll
do so much of this that this will just
be something that you'll manage to learn
but the second part of it which is
important is finding out the nature of
the stationary points
what type of stationary point this is we
used the second differential now in
order to find the second differential we
say that the derivative or the second
derivative must be greater than or equal
to zero if it is greater than or equal
to zero you get a minimum point if the
second differential is less than or
equal to zero you
a negative answer so that gives you a
maximum point and if the second
differential is equal to zero we'll deal
with us shortly but we call this a point
of inflection now this can often get
quite confusing so I like to think of it
as the second derivative being positive
so when you're positive you can imagine
that that's a smiley face so a smiley
face looks like a minimum point when you
sketch it out second differential being
negative is obviously you can think
about being an upside down curve a
negative face or a negative smile so
that gives you a maximum point and a
point of inflection is easy to remember
so that equals to zero so you will have
curves which you're dealing with which
have positives and negatives and maybe a
point of inflection at the same time and
you have to conclude or analyze those
graphs are those values fairly quickly
so it's important to find some faster
ways of working that out
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you