Global maximum

Definition

A global maximum is the highest point over the entire domain of a function. For polynomial functions, it is where the function attains its greatest value.

5 Must Know Facts For Your Next Test

1. The global maximum of a polynomial function can occur at a critical point or at the endpoints of a closed interval.
2. To find the global maximum, one must evaluate the function's value at critical points and endpoints (if applicable).
3. A polynomial with an even degree may have a global maximum if it opens downward.
4. The derivative's first test can help identify critical points where a global maximum might occur.
5. For polynomials of odd degree, there might not be any global maxima as they can extend to positive infinity.

Review Questions

• How do you determine if a given point is a global maximum for a polynomial function?
• Why might an odd-degree polynomial lack a global maximum?
• What role do critical points play in finding the global maximum?

Related terms

Critical Point: A point on the graph of a function where its derivative is zero or undefined.

Local Maximum: A point where the function reaches its highest value within a neighborhood around that point.

End Behavior: \text{The behavior of the graph of a polynomial function as } x \rightarrow \infty \text{ or } x \rightarrow -\infty.\