# Inflection Points

How to find the inflection points of the given graph: definitions, example, and its solution.

## Concave Up

If *f*''(*x*) is (+),*f*'(*x*), the slope of the tangent line, increases:

..., -3, -2, -1, 0, 1, 2, 3, ... .

Then *y* = *f*(*x*) is concave up. (= bend up)

If *f*'(*x*) is (-) and *f*''(*x*) is (+),

then *y* = *f*(*x*) is decreasing smoother and smoother.

If *f*'(*x*) and *f*''(*x*) are both (+),

then *y* = *f*(*x*) is increasing steeper and steeper.

## Concave Down

If *f*''(*x*) is (-),*f*'(*x*), the slope of the tangent line, decreases:

..., 3, 2, 1, 0, -1, -2, -3, ... .

Then *y* = *f*(*x*) is concave down. (= bend down)

If *f*'(*x*) is (+) and *f*''(*x*) is (-),

then *y* = *f*(*x*) is increasing smoother and smoother.

If *f*'(*x*) and *f*''(*x*) are both (-),

then *y* = *f*(*x*) is decreasing steeper and steeper.

## Second Derivative Test

The second derivative test is

to find the local maximum or the local minimum

by using *f*'(*x*) and *f*''(*x*).

If *f*'(*x*) = 0 and *f*''(*x*) is (-),

then that point is the local maximum point.

If *f*'(*x*) = 0 and *f*''(*x*) is (+),

then that point is the local minimum point.

## Inflection Point (Rising)

The rising inflection point is the point

where the sign of *f*''(*x*) changes

while *f*'(*x*) is (+).

So at the rising inflection point,*y* = *f*(*x*) either changes

1. from concave up to concave down

while increasing,

or

2. from concave down to concave up

while increasing.

(In this case,

the inflection point

is also a stationary point: *f*'(*x*) = 0.)

## Inflection Point (Falling)

The falling inflection point is the point

where the sign of *f*''(*x*) changes

while *f*'(*x*) is (-).

So at the falling inflection point,*y* = *f*(*x*) either changes

1. from concave up to concave down

while decreasing,

or

2. from concave down to concave up

while decreasing.

(In this case,

the inflection point

is also a stationary point: *f*'(*x*) = 0.)

## Example

Find the zeros of *f*'(*x*) = 0

Then *x* = 0, 2.

Derivatives of polynomials

Solving a quadratic equation by factoring

So the graph of *y* = *f*'(*x*) looks like this.

Quadratic function - factored form

Find the zero of *f*''(*x*) = 0

Then *x* = 1.

Second derivative

So the graph of *y* = *f*''(*x*) looks like this.

Point-slope form

Make a table like this.

Row 1:

Write the *x* values:

... , 0, ..., 1, ... , 2, and ... .

Row 2:

See the graph of *y* = *f*'(*x*)

and write the sign of *f*'(*x*).

If *x* = 0 or 2,

then *f*'(*x*) = 0.

If *x* < 0,

then *y* = *f*'(*x*) is above the *x*-axis.

So *f*'(*x*) is (+).

If 0 < *x* < 2,

then *y* = *f*'(*x*) is below the *x*-axis.

So *f*'(*x*) is (-).

And if *x* > 2,

then *y* = *f*'(*x*) is above the *x*-axis.

So *f*'(*x*) is (+).

Row 3:

See the graph of *y* = *f*''(*x*)

and write the sign of *f*''(*x*).

If *x* = 1,

then *f*''(*x*) = 0.

If *x* < 1,

then *y* = *f*''(*x*) is below the *x*-axis.

So *f*''(*x*) is (-).

If *x* > 1,

then *y* = *f*''(*x*) is above the *x*-axis.

So *f*''(*x*) is (+).

Row 4:

Mark how the graph of *y* = *f*(*x*) looks like.

If *f*'(*x*) is (+) and *f*''(*x*) is (-),

then *y* = *f*(*x*) is increasing smoother and smoother.

If *f*'(*x*) is (-) and *f*''(*x*) is (-),

then *y* = *f*(*x*) is decreasing steeper and steeper.

If *f*'(*x*) is (-) and *f*''(*x*) is (+),

then *y* = *f*(*x*) is decreasing smoother and smoother.

And if *f*'(*x*) is (+) and *f*''(*x*) is (+),

then *y* = *f*(*x*) is increasing steeper and steeper.

Then find *f*(0), *f*(1), and *f*(2).*f*(0) = 6*f*(1) = 4*f*(2) = 2

Fill in the blanks of the table.*f*(0) = 6, *f*(1) = 4, *f*(2) = 2

So the local maximum point is (0, 6).

The local minimum point is (2, 2),

Local maximum, local minimum

And the inflection point (falling) is (1, 4).

You can draw *y* = *f*(*x*)

by using the table above.

Starting from the left,

the graph goes up smoother and smoother until (0, 6),

the graph goes down steeper and steeper until (1, 4),

the graph goes down smoother and smoother until (2, 2),

then the graph goes up steeper and steeper.

As you can see,*y* = *f*(*x*) is concave down at the left side of (1, 1),

and *y* = *f*(*x*) is concave up at the right side of (1, 1).

So (1, 1) is the inflection point.