A close match
Problem
Two functions $f(x)$ and $g(x)$ were plotted on the same axes, where
$$
f(x) =\left(\frac{a}{x}\right)^x\quad \quad g(x) = b\exp\left(-\frac{(x-c)^2}{d}\right)
$$
I chose the coefficients $a, b, c$ and $d$ so as to make the function $g(x)$ match $f(x)$ 'as closely as possible' for points past the maximum of $f(x)$. My resulting charts were as follows.
Is it possible to approximately work out the values I chose? Can you choose values to obtain a closer match between the two?
Curve fitting/matching is big business in financial mathematics, where the goal is to be able to quote prices for non-liquidly traded products. To do this you need volatility and interest rate curves which are found by interpolating in some clever way between data points obtained from the price of standard (vanilla) traded products. For tractible mathematical analysis it often helps to try to base these curves in some sense on standard mathematical functions.
Student Solutions
I was delighted to receive this solution from Niharika Paul, one of our younger solvers.
The actual functions plotted were as follows:
Two functions $f(x)$ and $g(x)$ were plotted on the same axes, where
$$
f(x) =\left(\frac{20}{x}\right)^x\quad \quad g(x) = 1568\exp\left(-\frac{(x-7.3576)^2}{17.6232}\right)
$$
The coefficients in $g(x)$ were chosen so as to make the function $g(x)$ match $f(x)$ as closely as possible for points past the maximum of $f(x)$
Their charts at various points are