As I understand, it is an optimization problem.

LeastSquare=sum[y-f(x)]^2

Minimize least square by solving equations about a, b, and c. An iterative method could be developed to get the result or some R functions might be found useful. Please refer

http://r.789695.n4.nabble.com/Is-there-any-Gauss-newton-library-function-in-R-td897353.htmlHope it helps.