7A. Extreme Value Theorem

Instructions:

Use the input box for f(x) to define the function.

Extreme Value Theorem

Extreme values (maximum/minimum values) can be classified as local or global.
  • A maximum/minimum is local if it is only the largest/smallest value of f in a small viewing window around it. In other words, if you move far enough away from a local maximum/minimum you can find other values of the function that are larger/smaller.
  • A maximum/minimum is global if it is the largest/smallest value across the entire domain of f. In other words, a global maximum is the largest local maximum value and a global minimum is the smallest local minimum value.
The Extreme Value Theorem states that a function defined on a closed interval [a,b] is guaranteed to have both a global maximum and a global minimum. To find global extreme values:
  • Find all the critical points of f between the endpoints of the domain, a and b.
  • Evaluate f(x) at the critical points and endpoints. These are the only locations where f can have maximum or minimum values.
  • Because a global maximum/minimum is guaranteed to exist (by the theorem), the largest f(x) value gives the global maximum and the smallest f(x) value gives the global minimum.