yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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


3m read
·Nov 11, 2024

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.

More Articles

View All
Metaverse: Beyond Human
Imagine a world where you wake up, head to the office in the morning, to a party with friends in the evening, and then a live concert at midnight, all while sitting in the warmth of your home or from the comfort of your bed. That might just be part of hum…
$80,000,000 private jet tour!
Behind us is the future of corporate aviation. Let’s go inside and take a look. This is even bigger than the 6X. This is the standard bedroom with a king-size bed. But we have a larger one, which is a suite where you can have a desk, cabinet, and a seat …
Paying yourself first | Budgeting and saving | Financial Literacy | Khan Academy
You might have heard the term “paying yourself first,” and this just means putting your safety, your needs, especially your future needs, first before you think about other things. So let’s give ourselves an example. Let’s say that you want to buy a lapt…
How To Win A Business Pitch | Startup World Cup 2022
Foreign [Music] [Applause] [Music] Thank you so much for coming to start a World Cup. Well, everybody here, I think, well, most people have seen you on Shark Tank, and they know you’re into investments. But I want to start with how did you become an inves…
Geoff Ralston's Intro - Startup Investor School Day 1
Welcome everyone to my competitors’ startup investor school. If you think you’re at a different class, you should leave now. So it’s great to see you all here. I’m Jeff Ralston, and I’m going to act kind of as the master of ceremonies. I’ll be introducin…
Warren Buffett Buys GOLD?
Well, it’s that time again. The 13Fs are out. Uh, so we as the little investors get to have a look at what the big money managers of the world are buying and selling. And definitely the most watched 13F filing is definitely that of Mr. Warren Buffett. Uh…