Analyzing concavity (algebraic) | AP Calculus AB | Khan Academy

So I have the function G here; it's expressed as a fourth degree polynomial. I want to think about the intervals over which G is either concave upwards or concave downwards. Let's just remind ourselves what these things look like.

Concave upwards is an interval over which the slope is increasing. It tends to look like an upward-opening U. You can see here that the slope over here is negative, and then as X increases, it becomes less negative. It actually approaches zero, becomes zero, then crosses zero, and becomes slightly positive, then more positive, and even more positive. So you can see the slope is constantly increasing.

If you think about it in terms of derivatives, it means that your first derivative is increasing over that interval. In order for your first derivative to be increasing over that interval, your second derivative, let me write it as G because we're using G in this example, must be greater than zero.

Concave downward is the opposite; concave downward is an interval where you're going to be concave downward when your slope is decreasing. So G Prime of X is decreasing, or we can say that our second derivative is less than zero. Once again, I could draw it.

When X is lower, it looks like we have a positive slope, then it becomes less positive, and then it becomes less positive approaching zero. It becomes zero, then it becomes negative, then even more negative, and then even more negative. So, as you see, our slope is constantly decreasing as X increases here.

To think about the intervals where G is either concave upward or concave downward, what we need to do is find the second derivative of G and then think about the points at which the second derivative can go from positive to negative or negative to positive. Those will be places where it's either undefined or where the second derivative is equal to zero.

Then, let's see what's happening in the intervals between, and then we'll know over what intervals we are concave upward or concave downward. So let's do that. Let's first take the first derivative G Prime of X. I'm just going to apply the power rule a lot: 4 * -1 is -4, x^3 plus 2 * 6 is +12, x to the 1st power—I can just write it as x—and then minus 2. I could say -2 x to the zeroth power, but that's just -2, and the derivative of a constant is just zero.

Now, I can take the second derivative. G Prime Prime of X is going to be equal to 3 * -4, which is -12, x^2 (or decrement the exponent) plus 12. So let's see where this could be undefined. The second derivative is just a quadratic expression, which would be defined for any X, so it's not going to be undefined anywhere.

Interesting points where we could transition from negative to positive or positive to negative for the second derivative are where this could be equal to zero. Let's figure that out. So let's figure out where -12x + 12 could be equal to zero.

We could subtract 12 from both sides, and we get -12 x^2 is equal to 12. Dividing both sides by -12, we get x^2 is equal to 1. Either x could be equal to plus or minus or x could be equal to the square root of one, which is of course just one.

At this point, the second derivative at plus or minus one is equal to zero, so either between plus or minus one or on either side of them, we could be concave upward or concave downward. Let's think about this, and to do that, I'm going to make a number line. Let me find a nice soothing color here.

All right, that's a nice soothing color, and let's say, actually, make the number line a little bit bigger. If this is zero, this is 1, this is -2, this is positive 1, and this is positive 2, we know that at X = -1 and X = 1, our second derivative is equal to zero. Let's think about what's happening in between those places to see if our second derivative is positive or negative.

From this interval, we're going from negative Infinity to -1. Let's just try a value in that interval to see whether our second derivative is positive or negative. A good value could be -2, which is in that interval, so let's take G Prime Prime of -2.

This is equal to -2 * 4, because -2 squared is 4, so it’s -48 + 12, which is equal to -36. The important thing to realize is that if over here it’s negative, then over this whole interval, because it’s not crossing through zero (it's not continuous at any of these points), G Prime Prime of X is less than zero. This means that over this interval we are concave downward.

Now let's go to the interval between -1 and 1. This is the open interval between -1 and 1, and let's try a value there. Let’s try zero; that will be easy to compute G Prime Prime of 0. Well, when X is 0, this is zero, so it’s just going to be equal to 12.

The important thing to realize is our second derivative here is greater than zero, so we are concave upward on this interval between -1 and 1. Finally, let's look at the interval where X is greater than 1. This is the interval from one to Infinity.

If we want to view it that way, let’s just try a value; let’s try G Prime Prime of 2, because that's in the interval. G Prime Prime of 2 is going to be the same thing as G Prime Prime of -2, because whether you have -2 or 2, when squared, it becomes four. You are going to have 4 * -12, which is -48 + 12, which is -36.

So once again, on this interval you are concave downward. Now let's see if what we just established is consistent with what the graph actually looks like. We were able to come up with these insights about the concavity without graphing it, but now it’s kind of satisfying to take a look at a graph.

Let me see if I can match up the intervals. This is pretty closely matched right over here. Let me make it a little bit smaller. I am saying that I'm concave downward between negative Infinity all the way until negative one, all the way until this point right over here.

That looks right. It looks like the slope is constantly decreasing all the way until we get to X equal 1, and then the slope starts increasing. The slope starts increasing from there all the way until x equals 1, and then our slope starts decreasing again, and we get back into concave downwards.

What we were able to figure out by just taking the derivatives and doing a little algebra, we can see quite clearly looking at the graph.