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

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


3m read
·Nov 11, 2024

Let G of x = 1/4 x^4 - 4x^3 + 24x^2. For what values of x does the graph of G have an inflection point or have a point of inflection?

So, let's just remind ourselves what a point of inflection is. A point of inflection is where we change our concavity, or you could say where our second derivative G prime of x switches signs, switches which is signs.

So, let's study our second derivative. In order to study our second derivative, let's find it. So we know that G of x is equal to 1/4 x^4 - 4x^3 + 24x^2.

Given that, let's now find G prime of x. G prime of x is going to be equal to—I’m just going to apply the power rule multiple times. 4 * 1/4 is just 1—I'm not going to write the 1 down—it's going to be 1 * x^(4-1), or x^3. Then, -3 * 4 is -12, so that's -12x^(3-1), or -12x^2. Finally, 2 * 24 is 48, so that's 48x^(2-1), or 48x.

I could just write that as 48x. So there you have it, I have our first derivative.

Now we want to find our second derivative. So G prime prime of x is just the derivative of the first derivative with respect to x, applying the power rule again: 3x^2 - 24x + 48.

Let's think about where this switches signs. This is a continuous function; it's going to be defined for all x's. So, the only potential candidates for where it could switch signs are when this thing equals zero. So, let's see where it equals zero.

Let's set it equal to 0: 3x^2 - 24x + 48 = 0. Let's see, everything is divisible by three, so let's divide everything by three, we get x^2 - 8x + 16 = 0.

Can I factor this? Yeah, this factors to (x - 4)(x - 4), or you could just view this as (x - 4)^2 = 0, so x - 4 = 0, or x = 4.

Thus, G prime prime of 4 = 0. Now let's see what's happening on either side of that point to check if we're actually switching signs or not.

Let me draw a number line here, so this is at 2, 3, 4, 5, and I could keep going. We know that something interesting is happening right over here. G prime prime of 4 is equal to zero.

So, let's think about what the second derivative is when we are less than four.

Let me just try G prime prime of 0 since that'll be easy to evaluate. G prime prime of 0 is just going to be equal to 48. So when we are less than four, our second derivative G prime is greater than zero.

So we're actually going to be concave upwards over this interval to the left of four.

Now, let's think about the right of four. So let me evaluate G prime prime of 10. G prime prime of 10 is equal to 3 * 10^2, which is 300, minus 24 * 10, which is -240, plus 48.

So this is 60: 300 - 240 is 60 + 48. This is equal to 108, so it’s still positive.

So, on either side of four, G prime prime of x is greater than zero. So even though the second derivative at x = 4 is equal to zero, on either side we are concave upwards.

The second derivative is positive, and that was the only potential candidate.

There are no values of x for which G has a point of inflection. X = 4 would have been a value of x at which G had a point of inflection if the second derivative had switched signs here, going from positive to negative or negative to positive.

But it's just staying positive to positive. The second derivative is positive; it just touches zero right here and then goes positive again.

So going back to the question: for what x values does the graph of G have a point of inflection? No x values! I'll put an exclamation mark there just for drama!

More Articles

View All
Rick Astley - Never Gonna Give You Up (Official Music Video)
Rick Astley: [Music] We’re no strangers to love, you know the rules and so do I. I full commitments while I’m thinking of you, wouldn’t get this from any other guy. I just want to tell you how I’m feeling; got to make you understand. Never Going To Give Y…
Be Too Busy to “Do Coffee”
We squander our time with the death of a thousand cuts. So another tweet ahead was, “You should be too busy to do coffee while still keeping an uncluttered calendar.” People who know me know that I’m famous for simultaneously doing two things. One is havi…
Solving equations with zero product property
Let’s say that we’ve got the equation (2x - 1) times (x + 4) is equal to (0). Pause this video and see if you can figure out the (x) values that would satisfy this equation, or essentially our solutions to this equation. All right, now let’s work through …
Sea Turtles 101 | National Geographic
(Mellow music) - [Narrator] Sea turtles are ancient mariners. Present in all but Earth’s coldest oceans, these marine reptiles are well-adapted to a life on the move. (Dramatic music) Sea turtles have existed since the time of the dinosaurs. The earliest …
15 Decisions You WONT Regret 20 Years From Now
Hey there, my friend. Now, this is the second part of a video we did a couple of weeks ago where we talked about the decisions you will regret 20 years from now. Just like it’s hard to see how these bad decisions will play out in the long term, the revers…
How Many Photos Have Been Taken?
Hey, Vsauce. Michael here. In 1826, this became the very first photograph ever taken. And in 1992, this became the very first image ever uploaded to the web. But how many photographs have we all taken, altogether, throughout all of history? Well, 1000memo…