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Problem 78 Hard Difficulty

(a) Show that $ f(x) = x + r^x $ is one-to-one.
(b) What is the value of $ f^{-1}(1)? $
(c) Use the formula from Exercise 77(a) to find $ (f^{-1})'(1). $

Answer

(a)Please see proof.
(b) $$
f^{-1}(1)=0
$$
(c) $$
\frac{1}{2}
$$

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Video Transcript

in this problem, we're given a function of black sort of perfect explosive. Each X in front 81 short at this punch. One hole. Let's say, uh, look at first, they were job dysfunction. That final thanks. That is one plus X. Well, let's look at the second derivative. And this each of the eggs 37 10 would be the JD X, and you would get the same behavior for all fire order. There it is. So for any value of X, we know that each of the eggs will be positive, meaning that all the derivatives will be positive. So it means that dysfunction is increasing always since that is always increasing. Increasing this function is 12 or all right park years front. Yours or no, let's say that, um oh, um, that's is so f universe. Let's write it this time instead of f universe off X plus, each of eggs is equal. Thanks. We want an X plus, you two x to be one. If that is born, we see that X should zero, and that is the answer. So we consider a f in worse off one is unequalled zero. All right, import CVS, But English front or beginning used the question that we drive in problem 77 Soviet. Oh, that f inwards Prime off exit was born over f crime of immerse effects. So that is more over a probable and universal one in part. Maybe find FT merciful as zero. So that is warm or prime. Zero. What? Is that a problem? Zero. We know that prime effects from A is equal to one plus you to the ex that primary zero would be born boss Jiro And that is due so that the answer would be, um, dinner's prime one. Is it on over to?