y = x^3-6*x^2-3*x+1
Hello Everyone, Tom from everystepcalculus.com and everystepphysics.com. We’re going to do a problem on Concavity. And I’m gonna show you how my program work on that. Index 8 to get to my menu. We scroll down in the c section to Concavity. There is is there, letter D. I always have you start a graph on your paper. You put the x axis and y-axis with these. Keep up with graphing by hand, these functions. And I have a function here of the internet, for example. You have to press Alpha before you enter anything in these entry lines. Press Alpha. And the problem is X cubed minus six times x squared minus three times x plus one. I always show you what you’ve entered, you can check it and see if it’s correct. If it is, we press okay. I know we want Concavity, number 4. I’m going to press the number here, number four. Here’s Concavity. And I give the examples, here’s Concavity down and heres concave up. This is a valley, this is a mountain type situation. See, you take the original function. Take the first derivative here which is this. On your paper and the second derivative which is this. Six x minus twelve. We set 6 x minus twelve to zero to find what the x is at that which gives us critical number, there. And with respect to the second derivative and x equals two. So here we draw a number line here with the with two in the middle. Now we’re going to choose some number above two and some number below two. And and plug it into the second derivative here. I do that for you. I say X equals three. So at the second derivative x equals three, the answer is 6. And you’ll notice the 6 is positive. That means that its going to be concave up. And it’ll look like this on the graph. If we do a number less than 2 which is one. And plug it into the second derivative and we get minus 6. So, this is negative. So this is gonna be down. And I show you this and the next screen. So this one is down when you when you graph this function, it’s going to be down like this is going to be up like. That’s called Concavity. And that’s how you do it . Now on your paper, you must write negative infinity to two and two to positive infinity. And then you’ll be correct in your test or homework. Pretty neat, huh?