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

A function f of x
is plotted below. Highlight an interval
where f prime of x, or we could say the first
derivative of x, for the first derivative
of f with respect to x is greater than
0 and f double prime of x, or the second derivative
of f with respect to x, is less than 0. 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
0 right over here. And then the function
starts increasing again, all the way until this
point right over here. It hits 0. And then it goes, and the
function starts decreasing. Just this first constraint
right over here tells us it's going to be something in
this interval right over there. And then they say where
the second derivative is less than 0. So this means that
the slope itself, 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 will be 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 look 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
steep, less steep, less steep all the way to when the
slope gets back to 0. 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 the same thing where
our function is increasing, but it's increasing at a
slower and slower rate. So our function is increasing
in this whole region right over here, and we
see it's really steep here, that it's getting less
steep and less steep. And it's getting
closer and closer to 0, the slope of
the tangent line or the rate of increase
of the function. So I would pick anything right
around this region right here.

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