The extreme value theorem states that a continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. As shown in figure, one or both of these absolute extrema could occur at an endpoint. If youve ever ridden inside an elevator, frequented a movie theater, or lounged the day away at a public pool, then youve probably noticed a sign saying, maximum capacity. Gosh, even our smartphones have a limit on battery and storage capacity. But have you ever stopped to wonder what does this really means?
Find all extrema of the function on the boundary. This usually involves the calculus i approach for this work. The largest and smallest values found in the first two steps are the absolute minimum and the absolute maximum of the function. Here is the procedure for finding absolute extrema. Verify that the function is continuous on the interval [a,b] [a, b]. Find all critical points of f (x) f (x) that are in the interval [a,b] [a, b]. This makes sense if you think about it. The extreme value theorem states that a continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. As shown in figure \(\pageindex{2}\), one or both of these absolute extrema could occur at an endpoint. Next, place the critical numbers and the.
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