State the Bolzano-Weierstrass theorem. Use it to show that a continuous function attains a global maximum; that is, there is a real number such that for all .
A function is said to attain a local maximum at if there is some such that whenever . Suppose that is twice differentiable, and that for all . Show that there is at most one at which attains a local maximum.
If there is a constant such that for all , show that attains a global maximum. [Hint: if for all , then is decreasing.]
Must attain a global maximum if we merely require for all Justify your answer.