- #1
dsfranca
- 23
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Hi, I didn't understand what I should do exactly in this problem and I would really appreciate it if you guys could give me some directions!
Given the "bump function" B(x)=exp(-1/x²) for x>0 and 0 if x<=0, modify it to construct a C$^{\infty}$ function which satisfies:\begin{enumerate}<br /> \item C(x)=0 if $x\leq0$ <br /> \item C(x)=1 if $x\geq1$ <br /> \item C'(x)>0 if $0<x<1$<br /> \end{enumerate}<br />
I have proved in previous exercises that B$ ^{(n)'}(0)=0$ for all n.
My problem is I don't understand what is meant with "modify". As I have to solve many similar problems, it would be of great help if someone could point out what I should do.
Thanks!
Homework Statement
Given the "bump function" B(x)=exp(-1/x²) for x>0 and 0 if x<=0, modify it to construct a C$^{\infty}$ function which satisfies:\begin{enumerate}<br /> \item C(x)=0 if $x\leq0$ <br /> \item C(x)=1 if $x\geq1$ <br /> \item C'(x)>0 if $0<x<1$<br /> \end{enumerate}<br />
Homework Equations
I have proved in previous exercises that B$ ^{(n)'}(0)=0$ for all n.
The Attempt at a Solution
My problem is I don't understand what is meant with "modify". As I have to solve many similar problems, it would be of great help if someone could point out what I should do.
Thanks!
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