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.

Titus works at a hotel. part of his job is to keep the complimentary pitcher of water at least half full and always with ice. when he starts his shift, the water level shows 4 gallons, or 128 cups of water. as the shift progresses, he records the level of the water every 10 minutes. after 2 hours, he uses a regression calculator to compute an equation for the decrease in water. his equation is w –0.414t + 129.549, where t is the number of minutes and w is the level of water. according to the equation, after about how many minutes would the water level be less than or equal to 64 cups?