Inflection Points

Inflection Points

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

Concave Up

The graph shows concave up when f''(x) is (+). The slope of y = f(x) increases.

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

The graph shows concave down when f''(x) is (-). The slope of y = f(x) decreases.

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

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.

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 (+).

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 where the sign of f''(x) changes while f'(x) is (-).

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

f(x) is given below. f(x) = x^3 - 3x^2 + 6. 1. Find the inflection point(s) of f(x). 2. Find the extrema point(s) of f(x).

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.