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## Determining concavity of intervals and finding points of inflection: graphical

# Inflection points introduction

AP.CALC:

FUN‑4 (EU)

, FUN‑4.A (LO)

, FUN‑4.A.4 (EK)

, FUN‑4.A.5 (EK)

## Video transcript

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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