I am currently doing a problem that asks me to use the intermediate value theorem to show that x^(1/3)=1−x lies between 0 and 1. I want to start by evaluating the function at 0 and 1, but it seems the function is undefined because if you plug in 1, you get 1=0. This doesn't seem right. Is my interpretation correct?

Marcelo Maxwell

Marcelo Maxwell

Answered question

2022-09-23

I am currently doing a problem that asks me to use the intermediate value theorem to show that x 1 / 3 = 1 x lies between 0 and 1. I want to start by evaluating the function at 0 and 1, but it seems the function is undefined because if you plug in 1, you get 1 = 0. This doesn't seem right. Is my interpretation correct?

Answer & Explanation

embraci4i

embraci4i

Beginner2022-09-24Added 10 answers

Hint: Let f ( x ) = x 1 / 3 1 + x. Now f ( 0 ) = 1 and f ( 1 ) = 1.
Valentina Holland

Valentina Holland

Beginner2022-09-25Added 5 answers

You should think of the original equation as having a question mark: x 1 / 3 = ? 1 x or x 1 / 3 1 + x = ? 0. Then we define for simplicity only (no other reason) f ( x ) = x 1 / 3 1 + x and we ask: for which x is f ( x ) = 0 or f ( x ) = ? 0. Then we plug in two x's which are easy to evaluate and give different sign to f. If the notation f confuses you, you can ignore it. But keep in mind to interpret the given equation with a question mark. The exercise "asks" you to find x so that this equation is fulfilled. It does not say that it is true for every x!

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