00:01
Okay, so for this problem, we are given the derivative of a function, x equals 3x squared minus 2.
00:13
And it looks like the only part of this question that you had trouble with was determining the relative maximum and the relative minimum.
00:21
So in order to find the relative max and the relative min of a function, also known as a local maximum or a local minimum, first, we need to find the critical points, which can be found by setting the derivative equal to 0 or where it doesn't exist.
00:40
So critical points are where f prime of x either equal 0 or doesn't exist.
00:50
In this case, there's no place where it doesn't exist because we're not dividing by x anywhere.
00:56
But we can set this to zero and see if there's a number or number where we have critical points.
01:02
So let's assume f prime of x is equal to zero and see where we can go with this.
01:09
So zero equals 3x squared minus two.
01:13
We can add two to both sides and get two equals 3x squared, divide by three on both sides and get two thirds equals x squared.
01:24
And then we can take the square root of both sides.
01:27
And we're actually going to get both positive and negative square root of two -thirds could be equal to x.
01:36
And that's because if we look back at our original problem, whatever we're putting in for x, we're squaring it.
01:43
So it doesn't matter if it's positive or negative because as soon as it gets squared, it's going to become positive anyway.
01:50
So we actually have two possible critical values for x.
01:55
And once we found our critical values, what we can do is go ahead and set up like a number line with our critical values on it.
02:05
So we have negative square root of two -thirds, and we have positive square root of two -thirds.
02:14
It also might be a good idea just to put those in context, crunch those numbers in a calculator, just so you can kind of see what those numbers are.
02:23
So those are about negative 0 .47 and positive 0 .47 -ish.
02:35
And the reason we want to do that is because now that we have our critical points, in order to figure out if we have a max or a min anywhere, we're going to have to test points and see what's happening at those points...