Extreme Value Theorem Examples

Let f = tan(x). On which of the following intervals can we use the Extreme Value Theorem to conclude that f must attain a maximum and minimum value on that interval? 

• 
• [0, π]
• (0, π)
• 

The function f is discontinuous at  and at , (f(x) = tan(x) has vertical asymptotes there) therefore we can't use the Extreme Value Theorem on any intervals that include either of those values.

• : We can't use the Extreme Value Theorem because f is discontinuous on this interval.

• [0, π]:  We can't use the Extreme Value Theorem because f is discontinuous on this interval.

• (0, π): We can't use the Extreme Value Theorem because this interval is not closed, and f has a vertical asymptote on this interval.

•  : We can use the Extreme Value Theorem since the f is continuous on this interval and the interval is closed.