 In this video, we provide the solution to question number seven for practice exam number four for math 1210 In which case we're given the function f of x equals x to the fourth over four minus five x cubed over three plus three x squared plus 23 over two and we're asked on which intervals is the is the function f concave downward Well, if a function's concave downward that means that second derivative is actually negative So we need to compute the second derivative identify where this thing is negative Now if you are scared of these fractions don't worry turns out it's good They're gonna disappear very quickly. These numbers were chosen strategically for your sake. That's the test taker here We had the first compute the first derivative here. So by the usual power rule, we're gonna get x cubed minus 5x squared plus 6x the derivative of 23 halves since it's the constant Let's go to zero so we can see what the first derivative the fractions of all disappeared So that's the generous part of things and now when we calculate the second ribs because come cavities determined by the second derivative We'll do the we'll do the derivative again So we get 3x squared minus 10x plus 6 and so now we have this quadratic function We have to identify when this thing is equal to zero So we'll have to solve this quadratic equation We can do that by factoring complete the square or maybe we could do the quadratic formula If you try to factor if you take three times six Right, that's 18 factors of 18 that add to be negative 10 well three and six together are gonna give us of course nine That's not good enough. We could try nine and two, but that gives us 11 So we can't quite hit it There's not a quite a sweet spot and we are gonna have to use the quadratic formula in this situation to find these Potential points of inflection so by the quadratic formula, we're gonna get negative B which is 10 plus or minus the square root of B squared which is gonna be a hundred minus four ac so four times three times six We'll come back to that one all over two a which is gonna give you two times three which is a six so Continuing on we had three times Three times six. We said earlier was 18 four times that is gonna give us 72 So let me just erase these numbers right here We get 72 like so and so if we take 100 Minus 72 we end up with 28 so we get the square of 28 over six 26 our 28 is not a perfect square, which is why we couldn't factor it earlier But 28 can be fact as four times seven four is a perfect square. So we end up with 10 plus or minus two two times the square root of seven over six for which since everything's divisible by two We can simplify our critical numbers to be x equal excuse me our potential points of inflection as Five plus or minus the square of seven over three So, you know numbers like this seem to now make a lot more sense going on here So then we have to think about the sign right where you concave down if concave down when you're negative So in terms of a sign chart we need to look at the smaller number five minus the square of seven over three And we have to take the larger number five Plus the square of seven over three. We can use test values like zeros between What's the number bigger than that probably like a hundred bigger than that we use test points if we wanted to we put these It's the second derivative or I'm excuse the fact that I since I know the second derivative is quadratic It looks like a parabola the leading coefficient is a three So it's going to concave upward the parabola is I'm not saying that's what our function F is doing The the second derivatives graph would look something like this. So you're gonna see that it's above Below above so we see positive negative positive for our second derivative So in terms of the function, we're gonna be concave up concave down and concave upward So the interval we're looking for is right here We want all numbers between the two potential points of inflection as these are in fact points of inflection So we see that the correct answer would then be D