Worked example: Inflection points from first derivative | AP Calculus AB | Khan Academy

So we're told let G be a differentiable function defined over the closed interval from -6 to 6. 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 leftmost inflection point, inflection point in the graph of G?

So they want to know the x value of the inflection points in the graph of G. 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 leftmost 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 is a point where our second derivative is switching signs. It's going from positive to negative or negative to positive. So let's consider that first scenario:

If G is going from positive to negative, what's the first derivative doing? Well, remember the second derivative is the derivative of the first derivative. So where the second derivative is positive, that means that the first derivative is increasing. So if the second derivative is going from positive to negative, that means the first derivative is going from increasing to decreasing.

From increasing to decreasing and the function itself, well, when the second derivative is positive, it means that the slope is constantly increasing, and so that means we are concave upwards.

So, concave upwards to downwards 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.

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 the second derivative going from positive to negative. It's also the other way around, anytime the second derivative is switching signs. So it's also the situation where we're going from negative to positive, or where the first derivative is going from decreasing to increasing, decreasing to increasing.

Well, let's see, we are decreasing, decreasing, decreasing, and then we're increasing. All right, so it's right there. 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. If you look at the choices, if we want to answer the original question, well, the leftmost one is that x is equal to 3.

It's x = -3. x = -1 is indeed an x value where we have an inflection point. And let's see, x = 1 is one, and so is x = 4. So they actually listed all of these as inflection points, and they just wanted the leftmost one.