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Use a graphing calculator or computer to graph both the curve and its curvature function $\kappa(x)$ on the same screen. Is the graph of $\kappa$ what you would expect?

$$y=x^{4}-2 x^{2}$$

$k(x)=\frac{\left|12 x^{2}-4\right|}{\left[1+\left(4 x^{3}-4 x\right)^{2}\right]^{3 / 2}}$

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all right. In this problem, we want to be able to graph the function y equals X to the fourth power minus two times X squared and the curvature of that function on the same graph window. So that way we can compare the two and see if we can get an idea from the graph of why How to predict what the curvature graphite look like. So first thing we need to do is actually find function for the curvature. What I'm gonna do is I'm gonna use the formula in in the box in the top right corner. So I'm going to need the first and second derivatives of why, with respect to X. So what will get for the first derivative is for execute minus for X and for the second derivative get 12 times X squared, minus four. So then we know that the curvature Kappa of X going to be equal to the absolute value of why double prime. So 12 X squared, minus four over one plus Why? Prime squared, sir. Four x cubed minus for X squared. All raised Teoh the three house power. And we don't need to really simplify this very much since we're gonna plug it into the calculator. So first, what we'll do is we will look at the graph of why so what I'll do is and I have a t I 84 calculator. All go to my graph menu. Why one clear anything that's in in there. So in 11 I'm going to put my function for why? So we're gonna have X rays to the fourth power minus two x squared, and we're gonna go ahead and graph that I've already messed around with the window a little bit so you can see what works best for you. But I found it easiest to view when we have arrange from negative 5 to 5 on our X coordinates and from negative 2 to 8 on our white coordinates. Then I'm gonna click grass so you can see what that looks like. All right. And remember, when we're looking at this, that the curvature at a point is how quickly the graph is changing directions at that point. So if I look at the curve that we get in our in our graphing calculator, we've seen it. There are three inflection points where this graph is changing direction or turning around. So we might expect for the curvature to be greatest where those points happen. So what we could say is that we expect to see the greatest curvature. No, at the inflection points of why, In other words, if we're looking for the greatest curvature, that means that we would expect Kappa of X or curvature function to have ah, relative Maxima at those inflection points. And we can see if we look at a graph we put our, uh, ex And why ticks facing at just one So we can see that are inflection points occur at X equals negative 10 and one. Now, let's see if we're right. What I'm gonna do is I'm gonna go back to my y equals menu. I'm gonna go toe. Why, too? So that way we can leave our original function on the graph, and I'm going to use my left Cherokee to hover over that, uh, that how bar and I'm gonna click. Enter once and you can see that it's changed from a thin bar to a thick bar. So that's just gonna change our line style. So that way we can make sure we see the graph clearly in distinction from the graph of why So what we're gonna do is we're gonna put in our function for capita Vex. That's gonna be the absolute value, which I can find him there. My math numbers meant you and we're gonna have 12 x squared minus for that's in the numerator. We're gonna divide that by one plus the quantity four x cubed minus for X but squared clears off those parentheses and that denominators raised to the power of three over to. And then I'm gonna click graphic, and what we're gonna see is the graph of our curvature also show up. And if we look closely, we can see that does have three relative Maxima. And those are indeed at those points where we have inflection on why so we have relative Maxima at X equals negative one X equals zero and what so this is how we can think about the relationship between the graph and a curve, and the graph of that curves curvature by considering where those points of I haven't. But the greatest change in direction occur which correspond to the greatest curvature and relative maximum on our on a graph of capital fix