What is the proof for formula for mode- L + (fm - f(m-1))(2fm - f(m-1) - f(m+1))) * c

The mode M of a frequency distribution with class interval
c ,the modal frequency fm and frequencices  fm-1 and fm+1 of the preceding and
succeeding the modal class interval  is given by:

M = L +
{(fm -fm-1)/2fm- fm-1 - fm+1)}*c, where L is the lower limit of the class
interval.

The proof is in the assumption that the mode
corresponds to the variate value M for which the class interval c is divided in the
ratio of the weights of the frequencies  (fm+1 - fm):(fm -
fm-1).

Therefore M divides the class interval  (L , L+c) in
the ratio (fm+1-fm):(fm-m-1).

Therefore the mode M is
somewhere after  by   (fm- fm-1) (L+c -
L)/{(fm-fm-1))+(fm+1-fm).

Therefore M =  L + (fm- fm-1)
(L+c - L)/{(fm-fm-1))+(fm+1-fm)

M = L +(fm-fm-1)*c/{2fm =
fm-1 - fm+1}.

Hope this helps.