Inflection Points and Concavity Intuition

Understanding concave upwards and downwards portions of graphs and the relation to the derivative. Inflection point intuition

The idea of concativity
will show up a lot in

your calculus class.

Well, what I want to do in this
video is, one, show you what

concativity is, or what concave
upwards and what concave

downwards is, and to have an
intuition of what those mean,

and then discuss the ideas of
inflection points, which are

really just transition points
between being either concave

upwards and downwards or
between downwards and upwards.

So the two ideas that we'll
talk about, one is being

concave upwards-- I'll write
that in this column right

here, and maybe in pink I'll
show you what the concave

downwards looks like.

So a very simple way to think
about it is that concave

upwards is kind of a U shape.

That's why it's called
concave upwards.

It might look
something like this.

Let me draw some axes so
you know that I'm actually

graphing something and
not just drawing a U.

So maybe if that is my axis
right there, a concave

upwards graph would look
something like this.

And we do it in green.

It could look
something like this.

So this graph, it'll just
keep going as x goes

positive and negative.

Over the entire domain of this
function, this is my f of x

right here, it is concave
upwards, and you can see

it has this U shape.

And I'll discuss in a second
what implication that has for

its slope or what the slope is
doing, but this is just very

easy to recognize visually.

Similarly, let me
draw an axis again.

I want to show you what
concave downwards looks like.

Let me draw some
axes right there.

I'm just trying an
arbitrary function f of x.

It could be anywhere.

It doesn't have to be-- you
know, the bottom point doesn't

have to be in the first
quadrant like this.

It's just the general U
shape I wanted to show

you for concave upwards.

Concave downwards, you could
probably guess what it looks

like based on-- if this is
upwards, downwards is just

going to be opposite.

It might look
something like this.

It might look something like
this, so maybe this is some

other function g of x.

And notice, it looks
like an upside down U.

Over the entire domain
of this function, it

is concave downwards.

So you can look at it, and
immediately see that, oh, well,

if it's like a U, it's upwards.

If it's like an upside
down U, it's downwards.

But what does that actually
mean for the slope?

So to understand that, let's
think about what's happening

to the slope here.

So I'm just going
to do it visually.

So at that point, the slope of
the tangent line, or the slope

of the actual graph, or the
instantaneous slope, however

you want to view it-- let me
see how well I can draw it--

it'll look something like that,
and that's the tangent line.

Then as we increase x, so
that's at this x-value

right here, right?

At that x-value, the
slope looks like that.

It's fairly negative.

Now, as we increase x, maybe
we go to another point

right here, what happens?

The tangent line will
look like this.

It's still downward sloping,
but it's less downward

sloping now, right?

Here, it was a very
steep downward slope.

Now it's a less steep
downward slope.

And we can keep going if
we go to this point.

We're still downward sloping,
but even less steep.

And that keeps happening.

We keep getting less and less
downward sloping, or less steep

in the downward direction,
until we get to this minimum

point, and you've seen this
before, where our slope

actually gets to zero, and
it just keeps increasing.

It keeps increasing over here.

If we go to this point, now
we're going-- we have a

steep upward increase.

As we increase our x even
further, the slope at this

point right here, it's
increased even more.

It's a steeper upward
sloping curve.

So I want to make it very
clear, even though if we go to

this left-hand point here, we
had a very steep downward

sloping curve, and as we went
up here, we went upward sloping

the whole time as our x
increased, the whole time as

we increased x, what was
happening to the slope?

The slope was increasing.

So as we increased x,
the slope increased.

And this is the definition of
a concave upwards interval

or section of this curve.

In this case, it was
this entire curve.

I'll start showing you
functions that mix it

up in a little bit.

Over here, we were
very negative.

We became less negative.

Becoming less negative is the
same thing as increasing.

Even less negative over here.

Then we were completely flat.

Zero is a higher slope than
some negative slope here.

And then we became positive,
and more positive, and even

more positive, so the entire
time our slope is increasing.

Now, let's look at the
concave downward scenario.

Let's start with a relatively
low x, maybe right here.

Here we have a pretty high
slope, right, at this point?

We have a very steep
upward slope.

As x increases, so we go to
a less negative x, so x is

increasing, we still have an
upward slope, but

it's less upward.

So our slope has decreased.

And over here, it's still an
upward sloping curve, right?

It's going from the bottom
left to the top right,

but it's flatter.

It's flattening.

It's becoming less positive
until we get to this point.

I didn't draw the original
curve that well, where we get

to maybe a maximum point
because the slope goes to zero.

And as we increase now, our
slope becomes negative.

It becomes even more negative.

It becomes even more negative,
so as we kind of travel, as we

increase our x, as we increase
our x over this curve,

our slope decreases.

Slope is decreasing.

Slope decreases.

I want to make that very clear.

The slope was very positive.

This doesn't mean that
the slope is negative.

It's just saying that over the
entire interval, the slope

continues to decrease.

It goes from very positive,
to less positive, to less

positive, to zero, to slightly
negative, to more negative, to

very negative, so it just
decreases the whole time.

In concave upwards, it
increases the whole time.

So let me draw an example
that might have the

combination of the two.

Maybe I'll leave this
around just so we can

look at it for reference.

Let me draw an
axis right there.

Not the straightest line
in the world, but it'll

suit our purposes.

Let me just draw
some curve here.

So maybe I have something
that looks like that.

Maybe I have a curve that
looks something like that.

Now, over what interval of this
function's domain, maybe I'll

call it h of x, is this curve
concave upwards or

concave downwards?

Well, you can look at it just
from inspection, just from

that original definition.

I said that concave upwards is
when you're like a U, concave

downwards is when you're
like an upside down U.

So over here it looks like
a U from roughly this

point onwards, right?

We have this U shape.

We have this U shape and
maybe the curve just

keeps going on there.

So, from here onwards, we are
concave-- let me write it over

here-- from there onwards, we
are-- I don't know want to

do it in the same color--
we are concave upwards.

And then on the interval before
that, from here and before

this point right here, we
are concave downwards.

You can just recognize that.

And let's explore what's
actually happening

with the slope.

So over here, the tangent line
has a pretty high-- let me do

it in a different color-- over
here, the tangent line has a

pretty high positive slope.

Then it has a less positive.

Then at this maximum
point, it goes to zero.

Then it goes slightly
negative, even more negative,

even more negative.

So up until this point, the
slope is decreasing, which is

completely consistent with
what we said about being

concave downwards.

But then something interesting
happens at a point

right about there.

I'm not being very accurate or
precise right there, but right

about there, you see that the
slope of the tangent line

looks like that, right?

And that was more negative than
right here, which is more

negative than right there.

But over here, something
interesting happens.

All of a sudden, my slope
starts to increase again.

It starts to-- I
want to draw it.

Let me draw it as neatly as
I could possibly draw it.

So over here, the slope
look something like this.

So if we go backwards from
there, if we go into more

negative x's, the x's before
that had a less negative slope,

so the slope was decreasing.

But all of a sudden here,
when we go here, the

slope is slightly higher.

It's less negative there, then
it's even less negative there,

then it goes to zero here, then
it goes slightly positive,

and even more positive.

So up until this point in
are functions domain, our

slope was decreasing.

Our slope was decreasing
up until this point.

And then after that point,
our slope is increasing.

So this point, it seems like
we should call it something,

and we do call it something.

This point right here, where we
go from concave downwards to

concave upwards, and it would
actually be true if it was the

other way around, this is
called an inflection point.

And this would also be called
an inflection point if

it was the other way.

If we went from a curve-- let
me draw a curve that looks like

this-- that was concave upwards
and then it's concave

downwards, the point that we're
switching, the point at which

you go from one to the other,
essentially your rate of the

change of slope switches signs,
that is an inflection.

So this is also an
inflection point.

So what does this mean?

Hopefully, you understand
visually what concave upwards

and concave downwards means,
and an inflection point, but

what does this mean for
the second derivative?

Remember, the derivative is
the rate of change-- so

let me write this down.

So f of x, that's just
the function, right?

That's just the function.

That's what we've
been graphing.

f prime of x, that's the slope
of the function, slope of

function at any point x.

You've seen that
multiple times already.

What's the second derivative?

Well, you could view it
as the derivative of the

first derivative, right?

That's the slope of the
slope, or the rate of

change of the slope itself.

So what did we say
about concave upwards?

concave upwards is when the
slope is increasing, which

means that the rate of change
of the slope is positive.

Well, what that means is that
the second derivative, which is

the slope of the slope, f prime
of x, is going to be

greater than zero.

That's what concave upwards
just tells us, right?

Because if this is greater than
zero, then that means that the

rate of change of the
slope is positive.

That's what we said
was concave upwards.

Now, in concave downwards,
the slope is decreasing.

The rate of change of the
slope is decreasing,

or it's a negative.

So in that case, the
second derivative would

be less than zero.

And then in the situation when
we're switching, if over here--

let's look at this situation.

We're concave downwards.

We're concave downwards over
this interval right here.

That means that the second
derivative has to be less than

zero over this interval.

The rate of change of
the slope is negative.

It's decreasing.

So over this interval, our
second derivative is less than

zero, but then we go concave
upwards, so our second

derivative has to be greater
than zero over this interval,

which implies if our second
derivative is continuous, that

the second derivative had to be
equal to zero at that point.

I'll do that with some
examples in future videos.

Now, this condition by itself
isn't enough to say that

it's an inflection point.

In order for something to be an
inflection point, the second

derivative has to switch signs.

It has to go from being
negative in a concave downward

to positive in concave upwards,
or it has to go from positive

in concave upwards to negative
in concave downwards.

And obviously, in between,
it's going to switch.

If you're switching signs,
you're going to hit zero.

So when the second derivative
is zero, it might be an

inflection point, and then you
want to test around it to see

if it's actually switching
signs, and I'll show you

that in the next video.

But hopefully, you at least
have an intuitive sense of what

inflection points look like and
what the second derivative

is telling us.

If the slope is constantly
increasing, then the rate

of change of slope is
positive, then the second

derivative is positive.

If the slope is decreasing,
then the rate of change of

slope is negative, which
tells us that the second

derivative is negative.

And the point in which you're
switching from a positive

second derivative to a negative
one or a negative one to a

positive one, that is what is
called an inflection point.