Possible Cases

In general, all of the following possibilities can arise:

• f has no maximum in its domain I.
• f has a maximum at exactly one point in I.
• f has a maximum at several different points in I.

However even if f has a maximum at several different points, f can have only one maximum value. Because if f has a maximum at c₁ and also at c₂, then

f(c₁) ≥ f(c₂)

and

f(c₂) ≥ f(c₁),

and therefore f(c₁) and f(c₂) are equal.