(3) (LATEX question) Give a proof of the second part of the Extreme Value Theorem, 6.3.5, that is, if f : [a, b] → R is continuous then f is bounded below and attains a minimum.

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?

Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.

What is the purpose of the Intermediate Value Theorem?