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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).
I will first explain critical points theoretically. One type of critical point is a local minimum or maximum in the graph. The equation y = x2 has a local minimum at (0, 0). It is a point at which the curve is lower than the surrounding points on either side. In this case, the minimum is also absolute, because it is the lowest part of the whole curve. The second sort of critical point is an inflection point. This is a point in the graph where on one side, the slop is increasing, and on the other side, the slope is decreasing. This can happen whether the curve is increasing or decreasing. Look in the following graph at the first inflection point.