Let f : (0, 1) → R be a continuous function such that limx→0 f(x) = limx→1 f(x) = 0. Show that f achieves either an absolute minimum or an absolute maximum on (0, 1) (but perhaps not both).
Hint: A picture could help. One might also think "too bad the function isn’t defined and continuous in all of [0, 1], then we could apply the min-max theorem." However, isn’t that easily remedied?
Solution. I will give two solutions. The second solution, slightly modified, can used to prove that the same result holds if f : R → Ris continuous and limx→±[infinity] f(x) = 0.

What is 1/3x5 on anuber line what does it look like on a number line

An automobile assembly line operation has a scheduled mean completion time, μ, of 12 minutes. the standard deviation of completion times is 1.6 minutes. it is claimed that, under new management, the mean completion time has decreased. to test this claim, a random sample of 33 completion times under new management was taken. the sample had a mean of 11.2 minutes. assume that the population is normally distributed. can we support, at the 0.05 level of significance, the claim that the mean completion time has decreased under new management? assume that the standard deviation of completion times has not changed.

The function f(x)= -(x+5)(x+1) is down. what is the range of the function?