Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
f(x) = x^3 + 2x + k. Prove that every function of this family has exactly one real root
 2 years ago
 2 years ago
f(x) = x^3 + 2x + k. Prove that every function of this family has exactly one real root
 2 years ago
 2 years ago

This Question is Closed

him1618Best ResponseYou've already chosen the best response.0
the derivative of this function is olways positive and hence it has no real root, and so the function has only one real root
 2 years ago

aama100Best ResponseYou've already chosen the best response.0
thanks, but you didn't prove that the function has roots or not ?
 2 years ago

him1618Best ResponseYou've already chosen the best response.0
the very fact that the derivative is positive and its an odddegree fn shows it has atleast one root
 2 years ago

aama100Best ResponseYou've already chosen the best response.0
Answer: Let's assume that k represents any real number, let's say k = 3 f(x) = 3+2 x+x^3 Let's take two numbers : x = 30 and x = 12 Then, f(30) = 27057 < 0 f(12) = 1755 > 0 By the Intermediate Value Theorem, there exists a number "c" between 30 and 12 such that f(c) = 0. So, the equation has a real root. Now, let's suppose that there exists two roots w and q. Then, f(w) = f(q) = 0. However, using the Rolle's Theorem, f'(s) = 0 for some s in the set of numbers (w,q), but f'(x) = 2+3 x^2 > 0 for all x. So, it is impossible to have f'(s) = 0 for some c; in other words, there will be no maximum or minimum critical points, which means that there will be exactly one real root Therefore, we conclude that for any function of family f(x) = x^3 + 2x + k, where k is any real number, there exists exactly one real root
 2 years ago

polpakBest ResponseYou've already chosen the best response.1
If you can show that the derivative is positive for all values of x then the function is constantly increasing.
 2 years ago

aama100Best ResponseYou've already chosen the best response.0
I have proven it that way. What do you think ?
 2 years ago

polpakBest ResponseYou've already chosen the best response.1
that's basically what we were saying, but more long winded ;)
 2 years ago

him1618Best ResponseYou've already chosen the best response.0
thats extremely technical and purist but i think mine is a much easier and correct way...kudos 2 u for such theoristical capability
 2 years ago

aama100Best ResponseYou've already chosen the best response.0
I did some research then I put my answer together.. Thanks all
 2 years ago

him1618Best ResponseYou've already chosen the best response.0
yeah uve said the same thing in a completely abstract and meticulous maner..great
 2 years ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.