00:01
So in order to show that something has an inverse, it needs to be always increasing or always increasing or always decreasing.
00:12
Some people use a phrase strictly.
00:16
But as we're looking at this function, f of x defined as the integral from 1 to 2x of the square root of 16 plus t squared, t to the fourth, excuse me, me d t so what we can do to see if something's always increasing or decreasing is find the derivative of this and this would be my fundamental theorem of calculus where i can replace this 2x in for t so we're looking at the square root of 16 plus 2x to the fourth power but then you have to multiply because it's the chain rule times the derivative of 2x which would be 2 so the reality that we have is that the domain of this function, well, i guess it doesn't matter.
01:15
Because if you look at, and let me just go ahead and write that this is always positive, because no matter what you plug in for x, i guess i could say greater than or equal to zero, but it can never be negative.
01:28
It is for all values of x that once you take it to the fourth power, you're going to get a positive.
01:34
So i don't know if you want to have a phrase or because that, no matter what you plug in for x, that will be always positive.
01:45
And then if you multiply it by two, it's going to give you a positive anyway.
01:48
So that is your proof.
01:52
That's the derivative is always greater than or equal to zero than is always positive, which implies f is, well, the monotonic.
02:05
I believe is the calculus word that we use, proving that the inverse exists.
02:11
So i think i would circle that...