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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 29

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

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

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01:40

Using the derivative, veri…

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Use the derivative to help…

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In Exercises $29-34,$ grap…

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Suppose that $f$ is a func…

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The function $f(x)=x^{3}+a…

So if we want to show this is 1 to 1 by taking the derivative, I believe actually, in 29 that is how I show this was always was 1 to 1. But I'll go ahead and do it again. So we first just apply the derivative on either side. Uh, so that would give us f prime of X is equal to so we can distribute this across so d by d. X of two x minus d by D. X of three. So the derivative of two x is just to the derivative of three is zero. So we have we're f prime of X is too, which is always greater than zero. And again if we have where the derivative is always positive, that implies it is always increasing. So we proved that this is 1 to 1 by showing the derivative is always positive, which implies it is always increasing. Yeah, So maybe I should also say that implies 1 to 1

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