Q. 19 - Calculus

Question

Sketch a labeled graph of a function that fails to satisfy the hypothesis of the Extreme Value Theorem, and illustrate on your graph that the conclusion of the Extreme Value Theorem does not necessarily hold.

Step-by-Step Solution

Answer

A labeled graph of a function that fails to satisfy the hypothesis of the Extreme Value Theorem is

and

Step 1. Given Information.

The Extreme Value Theorem states that if f is continuous on a closed interval [a, b], then there exist values M and m in the interval [a, b] such that f(M) is the maximum value of f(x) on [a, b] and f(m) is the minimum value of f(x) on [a, b]. 

Step 2. Sketching a graph

In the following graph, there is no minimum value on the interval [a, b].

In the following graph, there is no maximum value on the interval [a,b].