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

Inflection points (graphical) | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

We're told let G be a differentiable function defined over the closed interval from 4 to 4. The graph of G is given right over here, given below. How many inflection points does the graph of G have?

So let's just remind ourselves what are inflection points. Inflection points are where we change concavity.

So we go from concave upwards to concave downwards or concave downwards to concave upwards.

Another way you could think about it is that we're going from our slope increasing to our slope decreasing, or the other way around. Any points where your slope goes from decreasing to increasing.

So let's think about that. As we start off right over here, at the extreme left, it seems like we have a very high slope. It's a very steep curve, and then it stays increasing, but it's getting less positive.

So it's getting a little bit flatter. Our slope is at a very high level, but it's decreasing, decreasing, decreasing. The slope is increasing, decreasing even more, it's even more.

Then it’s actually going to zero; our slope is zero, and then it becomes negative. So our slope is still decreasing, and then it's becoming more and more negative.

Then right around here, it looks like it starts becoming less negative, or it starts increasing. So our slope is increasing; it's really just becoming less and less negative.

Then it’s going close to zero, approaching zero. It looks like our slope is zero right over here, but then it looks like right over there our slope begins decreasing again.

So it looks like our slope is decreasing again; it’s becoming more and more negative. It seems like something interesting happened right over there; we had a transition point.

Then right around here, it looks like it starts; the slope starts increasing again. So it looks like the slope starts increasing; it's negative, but it's becoming less and less and less negative.

Then it becomes zero, and then it becomes positive, and then more and more and more and more positive. So, inflection points are where we go from slope increasing to slope decreasing, so concave upwards to concave downwards.

This was an inflection point, and also from slope decreasing to slope increasing. So that's slope decreasing to slope increasing, and this is also slope decreasing to slope increasing.

So how many inflection points does the graph of G have? We can see that we've on this graph, well, it has three over the interval that at least we can see.

More Articles

View All
Dua Lipa: The 60 Minutes Interview
Plenty of teenagers want to become pop stars, but few convince their parents to let them pack their bags and move to another country to try and make it big. That’s what Dua Lipa did when she was just 15 years old. She’d taken some singing lessons but didn…
Warren Buffett: Value Investing vs. Growth Investing
So if you tell me that you own a business that’s going to grow to the sky, and isn’t that wonderful? I don’t know whether it’s wonderful or not until I know what the economics are of that growth. How much you have to put in today, and how much you will re…
Elli Sharef at Female Founders Conference 2014
Awesome! Hi everyone! I’m Kat Metallic, YCS Director of Outreach. I’m so super thrilled to see you all here. The next speaker that we have is Le Chef. Le is the founder of Higher Art, a tool that makes finding a job and recruiting employees less painful. …
The Last Star in the Universe – Red Dwarfs Explained
One day the last star will die, and the universe will turn dark forever. It will probably be a red dwarf; a tiny kind of star. That’s also one of our best bets to find alien life, and might be the last home of humanity before the universe becomes uninhabi…
Trigonometry review
I want to do a quick overview of trigonometry and the aspects of trig functions that are important to us as electrical engineers. So this isn’t meant to be a full class on trigonometry. If you haven’t had this subject before, this is something that you ca…
Jamie Dimon: The $35 Trillion Dollar Storm Brewing in the US Economy
What you should worry about is the deficit. Today it is 7% of GDP. When Volcker was around and we had very high inflation, it was 3 and a half percent. The debt to GDP is 35% back then, 1982. It’s 100% today. The deficit is the biggest peacetime deficit w…