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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 37

$f^{\prime}(x)=\frac{2}{3 x^{2/3}} > 0$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Missouri State University

Baylor University

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So in exercise 37 I'm pretty sure I actually show this is 1 to 1 by taking the derivative. So you've already done that one? You've already actually seen the steps for that. Or at least if you watch the video, if you haven't, then I'll just go ahead and walk through what we did in there. So the first thing to take this derivative, we want to write this as a rational power, so x to the one third. And now, if we go ahead and take the derivative of this So this gives us f prime of X Z two. So we use power rule will be two thirds and then power rule says, Take the power, move it out front, subtract one off. Um, so it would be X to the negative two thirds, and then we can rewrite this in the following ways will be two thirds times one over and first. That would be one over X to the two thirds which we can write as that squared cubed root. Now, expert is always going to be positive. Uh, the cube root is something positive is also always positive. Multiply by something positive that means this whole thing is always going to be strictly larger than zero and notice how, even at zero, um, this would just go to positive infinity. So we actually don't even need to worry about that either. So it's always going to be increasing. So, uh, this implies that ffx is always increasing, which then implies a FedEx is 1 to 1.

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