Consider the function fx 12x5 45x4 200x3 4 fx has inflection...

Consider the function $$f(x) = 12x^5 + 45x^4 - 200x^3 + 4.$$ f(x) has inflection points at (reading from left to right) x = D, E, and F where D is and E is 0 and F is For each of the following intervals, tell whether f(x) is concave up or concave down. $$(-\infty, D):$$ Concave Down $$(D, E):$$ Concave Up $$(E, F):$$ Concave Down $$(F, \infty):$$ Concave Up

Transcript

00:01 For this problem we're given the function f of x equals 12 x to the fifth plus 45 x to the fourth minus 80 x cubed plus 4.

00:09 We're asked to find the inflection points.

00:10 It tells us already there's going to be three of them.

00:12 We're going to call them d, e, and f from smallest to largest.

00:16 And then we want to know the concavity on each of these intervals.

00:20 So to find your inflection points what you want to do is find your second derivative.

00:26 So the first derivative is the one that gives us critical points and intervals of increasing and decreasing.

00:34 So this problem doesn't have anything to do with that so we're just going to find it and move on.

00:39 So that's going to be 60 x to the fourth plus, we'll take 45 times 4 which is 180 x to the third minus 240 x squared and 4 is just 0.

00:57 And then i want to take the derivative again to get my second derivative.

01:02 So 240 x cubed plus 540 x squared minus 480 x.

01:17 And then we're just going to set this equal to 0 and solve to find our inflection points.

01:30 So you can do this any way you want to solve.

01:32 It looks like it'd be pretty easy to take out, let's see, we could take out like a 60 x it looks like and then we could factor this.

02:15 Actually it looks like it may be, it doesn't factor nicely.

02:20 So in the interest of time i'm just going to graph this really quick...

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