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First we need to choose an interval: Then we can say that a local maximum is the point where: The height of the function at a is greater than (or equal to) the height anywhere else in that interval. Or, more briefly: f(a) â¥ f(x) for all x in the interval. In other words, there is no height greater than f(a).
Consider the function f (x) = (x -1) 2, for. The only critical point is x =1. And the first or second derivative test will imply that x =1 is a local minimum. Looking at the graph (see below) we see that the right endpoint of the interval [0,3] is the global maximum. This leads us to introduce the new concept of endpoint extrema. Indeed, if c is an endpoint of the domain of f (x), then f (x) is said to have an endpoint maximum at c iff for all x in the domain close to c.