$1=1+\frac{1}{x^3}$ this equation has no solutions since $\frac{1}{x^3}$ can never equal 0 and is undefined when x=0. There fore P is false.

We need to show that there exists an $x$ for which $x = x^3+1$. This is equivalent to showing that $f(x)$ has a root, where $f(x) = x^3 - x + 1$. Now use the Intermediate Value Theorem.

You are watching: Is there a number that is exactly 5 more than its cube?

Question:

"Is there a number that is exactly one more than its cube?"(In my particular case, this was problem 51 from section 2.4 of *Single Variable Calculus Concepts and Contexts |4e* by James Stewart)

I was just assigned this problem in a homework assignment for my calculus class. I took a slightly different approach compared to

Théophile, although using the IVT here is likely the solution which most professors would look for (especially if you were asked this question on a quiz or exam where you aren"t allowed a graphing calculator).

You can verify that an $x$ value which satisfies these parameters exists by graphing $y=x$ and $y=x^3+1$ as individual functions. There is one intersection between these two graphs, that gives us the $x$ value which satisfies the equation $x=(x^3)+1$

You can then plug that $x$ value into the equation $x=(x^3)+1$ to check the answer.

*Note* this approach serves more as an aid in conceptualizing this particular problem, rearranging the equation $x=(x^3)+1$ and using the IVT to solve for the root is a more exact means to a solution. That said, I initially found it difficult to imagine this scenario, so I thought I"d share my approach in case anyone else here is struggling in the same way.

Cheers!

Share

Cite

Follow

edited Jan 18 "20 at 22:00

answered Jan 18 "20 at 21:31

TylerTyler

2155 bronze badges

$\endgroup$

2

Add a comment |

## Your Answer

Thanks for contributing an answer to sommos.netematics Stack Exchange!

Please be sure to*answer the question*. Provide details and share your research!

But *avoid* …

Use sommos.netJax to format equations. sommos.netJax reference.

To learn more, see our tips on writing great answers.

See more: What Is Better When You Break It Riddle, Riddle : What Is Better When You Break It

Draft saved

Draft discarded

### Sign up or log in

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Submit

### Post as a guest

Name

Email Required, but never shown

### Post as a guest

Name

Required, but never shown

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

## Not the answer you're looking for? Browse other questions tagged proof-verification or ask your own question.

Upcoming Events

Featured on Meta

Related

11

Are there more even numbers than odd numbers?

5

Prove that for every rational number z and every irrational number x, there exists a unique irrational number such that x+y=z

11

Prove that there is no smallest positive real number

1

Handling undefined case in uniqueness proof (How to Prove It, Velleman; 5.6, 2)

1

Is writing "However it is a general fact that..." a valid statement in a proof?

2

Prove that there are exactly two solutions to the equation $x^3 = x^2$.

2

Let $a$ be a positive number. Then there exists exactly one natural number $b$ such that $b\text{++} =a$.

0

Prove by contradiction that a real number that is less than every positive real number cannot be positive

2

Prove that there are exactly $\phi(p-1)$ primitive roots modulo a prime $p$

Hot Network Questions more hot questions

Question feed

Subscribe to RSS

Question feed To subscribe to this RSS feed, copy and paste this URL into your RSS reader.

sommos.netematics

Company

Stack Exchange Network

site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. rev2021.11.22.40798

sommos.netematics Stack Exchange works best with JavaScript enabled

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.