$1=1+\frac{1}{x^3}$ 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

Asking for help, clarification, or responding to other answers.Making statements based on opinion; back them up with references or personal experience.

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
Email

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.