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## Justifying properties of functions using the second derivative

# Worked example: Inflection points from first derivative

## Video transcript

- [Voiceover] So we're told let g be a differentiable function defined over the closed interval
from negative six to six. The graph of its derivative, so they're giving the
graphing the derivative of g, g prime is given below. So this isn't the graph of g. This is the graph of g prime. What is the x value of the
left-most inflection point, inflection point in the graph of g. So they want, they don't
want to know the x value of the inflection points
in the graph of g prime, in this graph. They want to know the inflection points, the x values of the inflection
points, in the graph of g. And we have to figure
out the left-most one. So, let me just make a little table here, to think about what is
happening at inflection points in our second derivative,
our first derivative, and our actual function. So, this is g prime, prime. This is g prime. And this is our actual, I
guess you could call it, the original function. So an inflection point are points where our second derivative
is switching sides. It's going from positive
negative or negative to positive. So, let's consider that first scenario. So g, so going from positive to negative. Positive, positive, to negative. So if g prime prime, if the second derivative's
going from positive to negative, what is the first derivative doing? Well, remember, the second derivative, is the derivative of the first derivative. So, where the second
derivative is positive, where the second derivative is positive, that means that the first
derivative is increasing. So, if second derivative's
going from positive to negative, that means first derivative is going from increasing to decreasing. From increasing to decreasing. And the function itself, well when the second
derivative is positive, we are going to be, that means, that means that the slope
is constantly increasing. And so that means that
we are concave upwards. So, concave upwards. Upwards to downwards. To concave, to concave downwards. But they've given us the graph of g prime. So let's focus on what are the points where g prime is going from
increasing to decreasing. So let's see. G prime is increasing,
increasing, increasing, increasing, increasing at a slower rate, and then it starts decreasing. So, right over there it's going from increasing to decreasing. So then it's decreasing,
decreasing, decreasing. Then it goes increasing,
increasing, increasing, increasing, and then decreasing again. So that's another point where we're going from increasing to decreasing. And those are the only ones that look like we're going
from increasing to decreasing. But we're not done yet. Because it's not just about going from the second derivative going
from positive to negative, it's also the other way around. Any time the second
derivative is switching signs. So, it's also the situation where we're going from
negative to positive. Or, for the first derivative
is going from decreasing, decreasing to increasing. Decreasing to increasing. Well let's see we're
decreasing, decreasing, decreasing, and then
we're increasing, alright. So it's right there. And then we're increasing, decreasing, decreasing, decreasing,
and then we're increasing. So right over there. So these are the inflection points that I've just figured out visually. So, if you look at the choices, if we want to answer
the original question, well the left-most one is it
x is equal to negative three? It's x equals negative three. X equals negative one is indeed a x value, where we have an inflection point. And let's see, x equals two is one, and so is x equals four. So they actually listed, all
of these are inflection points. And they just wanted the left-most one.