Define a function for a curve

Question

I am stuck with this problem longer than I'd like to admit. I wanna define a function which looks like this:

Graph It starts at $(0,0)$, then increases up to a maximum and then decreases, never reaching $0$.

I would like to change the rate of increase, the maximum point, and the rate of decrease.

What's the domain of $f$? Is $f$ continuous? Is it smooth? I think you need to provide more information — francescop21, Jul 05 '18 at 09:36
@MattiP. For certain values of $\lambda$, sure. However, different values of $\lambda$ seems to me to change the shape drastically from what is in the picture. — Arthur, Jul 05 '18 at 09:37
$f(x)=Mx^\alpha e^{-\beta x}$ satisfy all your requirements for every $\alpha >0$, $\beta >0$, $M >0$ — francescop21, Jul 05 '18 at 09:52

score 0 · Answer 1 · answered Jul 05 '18 at 09:47

0

You could start by taking two positive real numbers $a$ and $b$ and then define $f(x)=\dfrac{ax}{x^2+b}$ ($x\in\mathbb{R}^+$).

answered Jul 05 '18 at 09:47

score 0 · Answer 2 · answered Jul 05 '18 at 09:52

0

You could try:

$$f(x)=\frac{\log(x+1)}{x+1}$$

for $x\in[0,+\infty)$

answered Jul 05 '18 at 09:52

Jaroslaw Matlak

2 Answers2