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# Analyzing concavity (graphical)

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.4 (EK)
,
FUN‑4.A.5 (EK)

## Video transcript

a function f of X is plotted below highlight an interval where f prime of x where we could say the first derivative of X for the first derivative of F with respect to X is greater than zero and F double prime of X or the second derivative of F with respect to X is less than zero so let's think about what they're saying so we're looking for a place where the first derivative is greater than 0 that means that the slope of the tangent line is positive that means that the function is increasing over that interval so if we just think about it here over this whole region right over here the function is clearly decreasing then the slope becomes zero right over here and then the function starts increasing again all the way until this point right over here it hits zero and then it goes and the function starts decreasing so we're going to just this first constraint right over here tells us going to be something in this interval right over there and then they say where the second derivative is less than zero so this means that the slope itself whether it's whether it's positive or negative that it's actually decreasing we are going to be concave downwards right over here the slope itself it could be positive but it'll becoming less and less and less positive and so we're looking for a place where the slope is positive but it's becoming less and less and less positive if you looking over I'd over here the slope is positive but the slope is increasing it's getting steeper and steeper and steeper as we go and then all of a sudden it starts getting less steep less deep less deep less deep all the way to when the slope gets back to zero so if we want to select an interval it would be this interval right over here our slope is positive our function is clearly increasing but it is increasing at a lower and lower rate so I will select that right over there let's do one more example a function f of X is plotted below highlight an interval where f prime of X is greater than 0 so same thing where our function is increasing but it's increasing at a slower and slower rate so our function is increasing so our function is increasing in this whole region right over here and we see it's really steep here then it's getting less deep and less deep and it's getting closer and closer to zero though slope of the tangent line or the rate of increase of the function so I would pick anything right around this region right here