Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider the function $f (x) = xe^x - 2,$ we want to study the properties of $f (x)$ so that we can apply numerical methods to solve the equation $f (x) = 0$.

Which option is false ?

  1. the function, $f (x)$ is well defined and continuous for all $x$ in the interval $(0,2)$

  2. the function, $f (x)$ has no discontinuity and no singularities

  3. the function, $f '(x)$ is well defined and continuous for all $x$ in the interval $(0,2)$

  4. the function, $f '(x)$ has no discontinuity and no singularities

  5. all of the above

I do not understand the different options given. How do I know if the function is well defined?

share|improve this question
    
I take it assignment means homework, so I'll add that tag. Sorry if I misunderstood. –  1015 Jul 15 '13 at 12:35
    
"Well-defined", in this context, just means, if you plug in a value of $x$, you get a value of $f$. –  Gerry Myerson Jul 15 '13 at 12:39
1  
Given any $x\in (0,2)$, is there a number $f(x)$ defined without ambiguity? If so, the function is well-defined on $(0,2)$. –  1015 Jul 15 '13 at 12:39
    
Hi, no that is fine. It is homework yes but we have no study guide and I do not understand the material in the textbook –  Dee Jul 15 '13 at 12:42
add comment

1 Answer 1

up vote 1 down vote accepted

Hint 1: Here is a plot of $f(x)$ and $f'(x)$ over the range (do they look continuous over the range, see any singularities over $\mathbb{R}$). Can you make an analytical argument over the range $(0,2)$ and $\mathbb{R}$, respectively, regarding continuity and singularities? Are both continuous? Do either of them have singularities?

enter image description here

Hint 2: Here is a plot of $f(x) = 0$.

enter image description here

Hint 3: From hint 2, did you try solving $f(x) = 0$ analytically? Can you?

share|improve this answer
    
@Dee: Are you sure you wrote the function correctly in the problem? Regards –  Amzoti Jul 15 '13 at 12:52
    
Ok, looking at the graph, it seems like it is continuous. I'm not sure what singularity means, sorry. –  Dee Jul 15 '13 at 13:33
    
What if you had $1/(x-1)$. Where is the discontinuity? Where is the singularity (en.wikipedia.org/wiki/Mathematical_singularity)? –  Amzoti Jul 15 '13 at 13:50
    
@Dee: Did this resolve your issues? –  Amzoti Jul 15 '13 at 20:17
    
Great (gr)hints, great graphs! ;-) Needs a TU + –  amWhy Jul 16 '13 at 0:10
show 1 more comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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