Justifying properties of functions using the second derivative
Second derivative test
- [Voiceover] So what I want to do in this video is familiarize ourselves with the second derivative test and before I even get into the nitty-gritty of it, I really just want to get an intuitive feel for what the second derivative test is telling us. So let me just draw some axes here. So let's say that's my y-axis, let's say this is my x-axis and let's say I have a function that has a relative maximum value at X equals C. So let's say we have a situation that looks something like that and X equals C is right over, so that's the point C, F of C. So if I can draw a straighter dotted line. So that is X being equal to C and we visually see that we have a local maximum point there and we can use our calculus tools to think about what's going on there. Well one thing that we know, we know that the slope of the tangent line, at least the way I've drawn it right over here, is equal to zero. So we could say F prime of C is equal to zero and the other thing we can see is that we are concave downward in the neighborhood around X equals C. So notice our slope is constantly decreasing and since your slope, notice it's positive, it's less positive, even less positive, it goes to zero, then it becomes negative, more negative and even more negative. So we know that F prime prime, we know that F prime prime of C is less than zero and so I haven't done any deep mathematical proof here, but if I have a critical point at X equals C, so F prime of C is equal to zero, and we also see that the second derivative there is less than zero. Well intuitively this makes sense that we are at a maximum value and we could go the other way if we are at a local minimum point at X equals C or relative minimum point. So our first derivative should still be equal to zero 'cause our slope of a tangent line right over there is still zero. So F prime of C is equal to zero. But in this second situation, we are concave upwards. The slope is constantly increasing. We have an upward opening bowl and so here we have a relative minimum value or we could say our second derivative is greater than zero. Visually we see it's a relative minimum value and we can tell just looking at our derivatives, at least the way I've drawn it, first derivative is equal to zero and we are concave upwards. Second derivative is greater than zero. And so this intuition that we hopefully just built up is what the second derivative test tells us. So it says hey look, if we're dealing with some function F, let's say it's a twice differentiable function. So that means that over some interval. So that means that you could find its first and second derivatives are defined and so let's say there's some point, X equals C, where its first derivative is equal to zero, so the slope of the tangent line is equal to zero, and the derivative exists in a neighborhood around C and most of the functions we deal with, if it's differentiable at C, it tends to be differentiable in the neighborhood around C and then we also assume that second derivative exists is twice differentiable. Well then we might be dealing with a maximum point, we might be dealing with a minimum point, or we might not know what we're dealing with and it might be neither a minimum or a maximum point. But using the second derivative test, if we take the second derivative and if we see that the second derivative is indeed less than zero, then we have a relative maximum point. Then so this is a situation that we started with right up there. If our second derivative is greater than zero, then we are in this situation right here, we're concave upwards. Where the slope is zero, that's the bottom of the bowl. We have a relative minimum point and if our second derivative is zero, it's inconclusive. We don't know what is actually going on at that point. We can't make any strong statement. So with that out of the way, let's just do a quick example just to see if this has gelled. Let's say that I have some twice differentiable function H and let's say that I tell you that H of eight is equal to five, I tell you that H prime of eight is equal to zero, and I tell you that the second derivative at X equals eight is equal to negative four. So given this, can you tell me whether the point eight comma five, so the point eight comma five, is it a relative minimum, relative minimum, maximum point or not enough info? Not enough info or inconclusive? And like always, pause the video and see if you can figure it out. Well we're assuming it's twice differentiable and I think it's safe to assume that and for the sake of our problem we're gonna assume that the derivative exists in a neighborhood around X equals eight. So in this example, C is eight. So point eight five is definitely on the curve. The derivative is equal to zero. So we're dealing potentially with one of these scenarios and our second derivative is less than zero. Second derivative is less than zero. So this threw us. So the fact that the second derivative, so H prime prime of eight is less than zero, tells us that we fall into this situation right over here. So just with the information they've given us, we can say that at the point eight comma five we have a relative maximum value or that this is a relative maximum point for this. If somehow they told us the second derivative was zero, then we would say it's inconclusive. If they told us and that's all they told us and if they told us the second derivative is greater than zero, then we would be dealing with a relative minimum value at X equals eight.