Extreme Value Theorem

The Extreme Value Theorem (EVT) says:

If a function f is continuous on the closed interval ax b, then f has a global minimum and a global maximum on that interval.

In finding the optimal value of some function we look for a global minimum or maximum, depending on the problem. How do we know that one exists?

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1. Piece of a parabola

The first graph shows a piece of a parabola on a closed interval. Clearly there is a global minimum and maximum, shown by the two black dots

2. Piece of a sine curve

Select the second example from the drop down menu. This shows a sine curve with an interval set to a couple of cycles. There is clearly a global minimum and maximum, although in this case they aren't unique. EVT doesn't guarantee uniqueness of global extrema, just that at least one minimum and one maximum will exist.

3. Open interval

Select the third example, showing the same piece of a parabola as the first example, only with an open interval. Since the endpoints are not included, they can't be the global extrema, and this interval has no global minimum or maximum. Hence Extreme Value Theorem requires a closed interval to avoid this problem

4. Discontinuous

Select the fourth example, showing an interval of a hyperbola with a vertical asymptote. There is no global extrema on this interval, which is a reason why the Extreme Value Theorem requires a continuous interval.

Differnt type of discontinuity

Select the fifth example, showing a different type of discontinuity. Here, there is no global maximum, showing again why the Extreme Value Theorem requires continuity on the interval.

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