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Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.Exercise 32

$f^{\prime}(x)=\frac{-1}{\sqrt{6}-2 x} < 0$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Campbell University

Harvey Mudd College

Baylor University

Idaho State University

Lectures

01:04

Using the derivative, veri…

01:40

01:55

02:09

00:56

Use the derivative to help…

01:25

01:11

01:17

In Exercises $29-34,$ grap…

01:45

01:18

The function $f(x)=x^{3}+a…

01:32

Determine whether the func…

So if we want to show that this is 1 to 1, they tell us to do it by showing the derivative is always increasing or decreasing. And I believe, actually end number 32. Uh, I actually do that to show this, but let's just go ahead and take the derivative. You haven't done that one yet, so we do deep I d. X from this. So that would be f prime of X. So now this to take the derivative. Remember, it's really to a one half power, so we use powerful first two. It's one half raised, the sixth minus two X now to the negative one half. But then we have to take the derivative of the inside due to chain rule. And then the derivative of this is just going to be negative, too. So those two's cancel and we end up with some negative one over this negative power makes it turn into our reciprocates it. And then the one half hours still square root so we'd be screwed of six minus two x. So now the square root of six minus two X is always going to be positive. And if we divide something always positive or I should say, always positive or zero uh into something that is negative. Then that tells us we would end up with a negative number. So this implies that couple backs is always decreasing. And if we have something always decreasing, this implies that ffx is 1 to 1.

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