Determining concavity of intervals and finding points of inflection: graphical
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Inflection points introduction
If you were paying close attention in the last video, an interesting question might have popped up in your brain. We have talked about the intervals over which the function is concave downwards. And then we talked about the interval over which the function is concave upwards. But we see here that there's a point at which we transition from being concave downwards to concave upwards. Before that point, the slope was decreasing, and then the slope starts increasing. The slope was decreasing and then the slope started increasing. So that's one way to look at it. Right here in our function we go from being concave downwards to concave upwards. When you look at our derivative at that point, our derivative went from decreasing to increasing. And when you look at our second derivative at that point, it went from being negative to positive. So this must have some type of a special name you're probably thinking. And you'd be thinking correctly. This point at which we transition from being concave downwards to concave upwards, or the point at which our derivative has a extrema point, or the point at which our second derivative switches signs like this, we call it an inflection point. And the most typical way that people think about how could you test for an inflection point, it's a point, well, conceptually, it's where you go from being a downward opening u to an upward opening u. Or when you go from being concave downwards to concave upwards. But the easiest test is it's a point at which your second derivative switches signs. So in this case, we went from negative to positive. But we could have also switched from being positive to negative. So inflection point, your second derivative f prime prime of x switches signs. Goes from being positive to negative or negative to positive. Switches signs. So this is a case where we went from concave downwards to concave upwards. If we went from concave upwards to concave downwards, like that, then this inflection point up until that point, the slope was increasing. So the second derivative would to be positive. And then the slope is decreasing, so your second derivative would be negative. So here your second derivative is going from positive to negative. Here your second derivative is going from negative to positive. In either case, you are talking about an inflection point.
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