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(a) Given $f(x)=2 x^{3}+3 x-4,$ show this function is one-to-one(b) Determine $\left(f^{-1}(x)\right)^{\prime}(18)$.

(a) $f^{\prime}(x)=6 x^{2}+3 > 0$(b) $1 / 27$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

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Ah. So first, if we want to show that this function is 1 to 1, we could try to graph it. But let me graph in. That might be kind of hard, so I'm just going to take the derivative of it and then show that this is always increasing or decreasing. So first, to take the derivative this we can use power rule. So just move three outfront, subtract off one So that give us six x squared. The derivative of three X would just be three and then 40 So X square is always greater than equal zero. Multiply that by six, still the same thing. And if we add three to this, this is going to be greater than equal to three, which is strictly larger than zero. So this implies that affects is always increasing, and that further implies that f of X is 1 to 1. Well, let me rewrite that. So f of X is 1 to 1. So now that we have that, if we want to find the directive of the inverse at 18, we can go ahead and use this formula down here. So we're just gonna plug 18 into it will be one over f in her purse for F prime of f inverse of 18. Um, but we need to figure out what it is f inverse of 18, and finding the actual inverse of this would actually be kind of hard to do. So what I'm going to do instead is just set this equal to 18 and then see if I can solve for this. Um, so you could just plug this into some equation solver and we'll split the answer out. But I'm just going to do it by hand. Zona first attract the 18 overs that gives to execute plus three x minus 22 +020 And now, um, you could maybe just try to guess a couple of numbers, But you could also use the rational roots here. Um, which says possible factors are going to be 22 the constant over our bleeding term. So that would be 1 22. Or actually, I should be saying plus or minus 1/22 uh, to 11 and then just wanted to. And then, uh, just so we don't have to watch me sit here and try to figure out what it is. The solution to this distance of being X is equal to two. So what this actually implies is that f inverse of 18. Secret of two. So we come over here, replace that with two. So we have one over f prime of two and we actually already have f prime right here. So we can just take to implode that in so f prime of X is equal to six x squared plus three is what we want f prime of two. Yes, that would be six times four plus three, which would be 24 plus three or 27. So this here ends up just being 1/27 so that will be our solution.

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