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(a) Given $f(x)=3 x^{5}+2 x^{3}+2,$ show this function is one-to-one.(b) Determine $\left(f^{-1}(x)\right)^{\prime}(7)$.

(a) $f^{\prime}(x)=15 x^{4}+6 x^{2} > 0$(b) $1 / 21$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Campbell University

McMaster University

University of Michigan - Ann Arbor

Lectures

03:13

(a) Given $f(x)=2 x^{3}+3 …

00:23

Given the one-to-one funct…

00:22

07:16

Find the indicated functio…

02:57

Find the function values.<…

00:30

11:19

Find the function $f(x)=a…

02:26

Show that the function $f(…

02:05

So if we first want to show this function is 1 to 1, we could try to graph it. But honestly, we found just using a calculator. I have no idea how to do that. So what we can do instead is take the derivative and hope this will always be increasing or decreasing. So this is going to give Prime of X is equal to So we use powerful here to move the five outfront subtract off one. It would be 15 x to the fourth, um, power rule for this as well. So that would be plus six x squared. And then that would just be zero so x to the fourth. Next quarter is always going to be positive. And then if we multiply those by positive numbers and ahead, that will also always be positive. So that's strictly greater than zero. Which implies FX is always increasing, which further implies ffx is 1 to 1. Okay, now that we have that we can come down here and so this says, Well, we want to find the derivative of the inverse at seven, and in order for us to do that, we can just follow this equation down here. So we're just going to replace this X with us seven. So this would be one over f prime of f inverse of seven. So we need to figure out what is f inverse of seven slots so that in this corner here, well, that is the same thing as asking. Well, what is the solution to three is eager to three X to the fifth plus two x cubed plus two. Um, so you could just blow this into an equation solver, um, to do it. But I'm just gonna go ahead and show you how you can also do this by hand. So first, we're going to go ahead and subtract seven over, so it gives us zero is 0 to 3 x, the fifth plus to execute minus five. And then, um maybe you can just guess what number this is. Um, but if you can't, you can go ahead and use the rational root for him to kind of help you figure out some numbers. So we look at the factors of our constant so five over the factors of our leading coefficient three. And so that's just going to be plus or minus one in +51 in three. And if you were to just, like, run through all of these, you would end up finding that X is just going to be equal to one. Uh, yeah, This one is actually not too hard to actually show this. Um, but what this actually implies, how is I? F in verse of seven is equal to one so that we could just come over here, replace that with one, and then this would be one over f prime of one. So also over here on the side, fair or what have prime of one is so f prime of X. Let me write this out again. Would be 15 x to the fourth plus six x squared. So if we plug in one, that would just be 15 or six, which is 21. So now we just go over here and plug in 21. So that would give us 1/21. So that would be our answer for part B.

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