Global Maximum, Global Minimum

Global Maximum, Global Minimum

How to find the global maximum and the global minimum of the given graph: examples and their solutions.

Example 1

Find the global extrema points of f(x). f(x) = x^3 - 3x + 1, [0, 2]

The global exterma means
either the global maximum or the global minimum
in the given interval.

So this example says
find the global maximum and the global minimum
in [0, 2].

Graphing intervals on a number line

Find the zeros of f'(x) = 0

Then x = -1, 1.

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: 0, ... , 1, ... , 2.

The given interval is [0, 2].
So just focus on this interval.

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

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

If 0 ≤ x < 1,
then y = f'(x) is below the x-axis.
So f'(x) is (-).

And if 1 < x ≤ 2,
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 = 1,
y = f(x) looks like ↘ ↗.
So x = 1 is the local minimum.

Local maximum, local minimum

Then find f(0), f(1), and f(2).

f(0) = 1
f(1) = -1
f(2) = 3

Fill in the blanks of the table.

f(0) = 1, f(1) = -1, f(2) = 3

The global maximum point has the greatest y value.
So the global maximum point is (2, 3).

The global minimum point has the least y value.
So the global minimum point is (1, -1).

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

Starting from (0, 1),
the graph goes down until (1, -1),
then the graph goes up until (2, 3).

So the global maximum point is (2, 3).
And the global minimum point is (1, -1).

Example 2

Find the global extrema points of f(x). f(x) = -x^4 + 2x^2 + 3, [0, 2]

Find the zeros of f'(x) = 0

Then x = 0, -1, 1.

Derivatives of polynomials

Solving polynomial equations

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

Graphing polynomial functions - Example 3

Make a table like this.

Row 1:
Write the x values: 0, ... , 1, ... , 2.

The given interval is [0, 2].
So just focus on this interval.

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

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

If 0 < x < 1,
then y = f'(x) is above the x-axis.
So f'(x) is (+).

And if 1 < x ≤ 2,
then y = f'(x) is below 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 ↗.

And 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.

Local maximum, local minimum

Then find f(0), f(1), and f(2).

f(0) = 3
f(1) = 4
f(2) = -5

Fill in the blanks of the table.

f(0) = 3, f(1) = 4, f(2) = -5

The global maximum point has the greatest y value.
So the global maximum point is (1, 4).

The global minimum point has the least y value.
So the global minimum point is (2, -5).

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

Starting from (0, 3),
the graph goes up until (1, 4),
then the graph goes down until (2, -5).

So the global maximum point is (1, 4).
And the global minimum point is (2, -5).