# Global Maximum, Global Minimum

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

## Example 1

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 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).