Extreme Values

Suppose , and . Minimum and maximum values of are called the extreme values.

Maximum

Maximum of the function is where iff:

aka. Global Maximum. Maximum doesn’t exist always.

Local Maximum

A Local maximum of the function is where iff:

Global maximum is obviously a local maximum.

The above statement can be simplified when or .

When :

When :

Minimum

Minimum of the function is where iff:

aka. Global Minimum. Minimum doesn’t exist always.

Local Minimum

Global minimum is obviously a local maximum.

The above statement can be simplified when or .

When :

When :

Special cases

f is continuous

Then by Extreme Value Theorem, we know has a minimum and maximum in .

f is differentiable

• If is a local maximum:
• If is a local maximum:
• and If is a local maximum:
• If is a local minimum:
• If is a local minimum:
• and If is a local minimum:

Critical point

is called a critical point iff: