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# 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 in up in your brain we have talked about the intervals over which the function is concave downwards 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 would be thinking correctly this this point at which we transition from being concave downwards to concave upwards or the point at which our derivative has an extremum point or the point at which we go at which our second derivative switches signs like this we call it an inflection point in in flexion in flexion 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 you to an upward-opening you or when you go from being concave downwards to concave upwards but the easiest test is it's a point at which your second derivatives which is sine so in this case we went from negative to positive but we could have also switched from being positive to negative so inflection point inflection point your second derivative F prime prime of X switches signs goes from being positive to negative or negative to positive which is sines so this is a case where we went from concave downwards to concave upwards if we went from going if we went from concave upwards to concave downwards to concave downwards like that then this inflection point at this inflection point up until that point the slope was increasing so the second relative would be positive and then the slope is decreasing the derivative is decreasing so your second derivative would be negative so here your second derivative is going from positive to negative here your second derivatives going from negative to positive in either case in either case you are talking about an inflection point
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