Local Maximum, Local Minimum

Local Maximum, Local Minimum

How to find the local maximum and the local minimum of the given graph: definitions, examples, and their solutions.

First Derivative Test

If f'(x) is (+), then y = f(x) increases. If f'(x) is (-), then y = f(x) decreases.

If f'(x) is (+),
the slope of the tangent line is (+).

Then y = f(x) increases. (↗)

If f'(x) is (-),
the slope of the tangent line is (-).

Then y = f(x) decreases. (↘)

Local Maximum, Local Minimum

The local maximum is the point where the sign of f'(x) changes from (+) to (-). The local minimum is the point where the sign of f'(x) changes from (-) to (+).

The local maximum is the highest point
near that point.

It's the point
where the sign of f'(x) changes from (+) to (-).

So, in most cases,
f'(a) = 0 can be the local maximum point.

f(x) changes from ↗ to ↘.

Stationary Point: the point where f'(a) = 0.

The local minimum is the lowest point
near that point.

It's the point
where the sign of f'(x) changes from (-) to (+).

So, in most cases,
f'(a) = 0 can be the local minimum point.

f(x) changes from ↘ to ↗.

The local maximum or the local minimum can be found even if f'(a) does not exist.

But, even if f'(a) does not exist,
(= not differentiable at x = a)
the local maximum or the local minimum can exist.

If the sign of f'(x) changes from (+) to (-), or vice versa,
then that point is the local maximum or the local minimum.

So the local maximum or the local minimum
don't have to be stationary points [f'(a) = 0].

Example 1

Find the local extrema point(s) of f(x). f(x) = x^3 - 3x^2 - 9x + 7

The local exterma means
either the local maximum or the local minimum.

So this example says
find the local maximum and the local minimum
if they exist.

Find the zeros of f'(x) = 0

Then x = -1, 3.

Derivatives of polynomials

Solving a quadratic equation by factoring

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

Quadratic function - factored form

Make a table like this.

Row 1:
Write the x values: ... , -1, ... , 3, and ... .

Row 2:
See the graph of y = f'(x)
and write the sign of f'(x).

If x = -1 or 3,
then f'(x) = 0.

If x < -1,
then y = f'(x) is above the x-axis.
So f'(x) is (+).

If -1 < x < 3,
then y = f'(x) is below the x-axis.
So f'(x) is (-).

And if x > 3,
then y = f'(x) is above the x-axis.
So f'(x) is (+).

Row 3:
Mark how the graph of y = f(x) looks like.

If f'(x) is (+),
then y = f(x) is increasing.
So mark ↗.

If f'(x) is (-),
then y = f(x) is decreasing.
So mark ↘.

Near x = -1,
y = f(x) looks like ↗ ↘.
So x = -1 is the local maximum.

And near x = 3,
y = f(x) looks like ↘ ↗.
So x = 3 is the local minimum.

So find f(-1) and f(3).

f(-1) = 12
f(3) = -20

Fill in the blanks of the table.

f(-1) = 12, f(3) = -20

So the local maximum point is (-1, 12).
And the local minimum point is (3, -20).

You can draw y = f(x)
by using the table above.

Starting from the left,
the graph goes up until (-1, 12),
the graph goes down until (3, -20),
then the graph goes up.

Example 2

Find the local extrema point(s) of f(x). f(x) = x^4 - 4x^3 + 10

Find the zeros of f'(x) = 0

Then x = 0, 3.

Derivatives of polynomials

Solving polynomial equations

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

Graphing polynomial functions

Make a table like this.

Row 1:
Write the x values: ... , 0, ... , 3, and ... .

Row 2:
See the graph of y = f'(x)
and write the sign of f'(x).

If x = 0 or 3,
then f'(x) = 0.

If x < 0 and 0 < x < 3,
then y = f'(x) is below the x-axis.
So f'(x) is (-).

And if x > 3,
then y = f'(x) is above the x-axis.
So f'(x) is (+).

Row 3:
Mark how the graph of y = f(x) looks like.

If f'(x) is (-),
then y = f(x) is decreasing.
So mark ↘.

If f'(x) is (+),
then y = f(x) is increasing.
So mark ↗.

Near x = 0,
y = f(x) looks like ↘ ↘.
So x = 0 is
neither the local maximum nor the local minimum.
(although f'(0) = 0)

Near x = 3,
y = f(x) looks like ↘ ↗.
So x = 3 is the local minimum.

So find f(3).

f(3) = -17

Fill in the blank of the table.

f(3) = -17

So the local minimum point is (3, -17).

You can draw y = f(x)
by using the table above.

Starting from the left,
the graph goes down,
passes through (0, f(0)) horizontally, [f'(0) = 0]
still goes down until (3, -17),
then the graph goes up.