00:01
Okay, so we have some function f, and we know the derivative of x.
00:04
We have the x prime of x is equal to x squared minus 2 over x.
00:08
And we want to find the interval on which f is decreasing.
00:11
So remember, a function is decreasing if its derivative is less than a equal to zero.
00:16
So we need to find x such that f prime of x is less than or equal to zero.
00:24
But we know what f prime of x is.
00:26
It's this.
00:26
So we need to solve x squared minus two over x, less than equal to.
00:31
To 0.
00:32
If we just rearrange slightly, this is the same as x squared is less than equal to 2 over x.
00:38
Okay, so now the idea is we want to multiply through by x, so we can bring this x up here.
00:43
We have to be a bit careful because if x is negative, then multiplying by x flips the inequality sign, so we need to take cases.
00:51
So case 1, if x is greater than 0, or if this case, we can just multiply both sides by x, and the inequality stays the same.
01:02
So we get x cubed is less than equal to two.
01:07
And taking cube roots of both sides, we get that x is less than or equal to the cube roots of two...