Use the Intermediate Value Theorem to show that the equation

hard12bb30crg

hard12bb30crg

Answered question

2022-05-10

Use the Intermediate Value Theorem to show that the equation x 3 + x + 1 = 0 has a solution.
How to do this?

Answer & Explanation

aitantiskbx2v

aitantiskbx2v

Beginner2022-05-11Added 16 answers

Let f ( x ) = x 3 + x + 1. Then f is a polynomial, so its continuous. Then we notice that
f ( 1 ) = 1 ,  and  f ( 0 ) = 1.
But then by the intermediate value theorem, this means that for all y [ 1 , 1 ] we have some x [ 1 , 0 ] such that f ( x ) = y. Therefore we have some z such that f ( z ) = 0.

Since f ( z ) = 0 we know that z 3 + z + 1 = 0 and so the equation
x 3 + x + 1 = 0
has a solution, namely z.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?