# cnansari fu'ivla

x1 is the mean-value theorem mean/forward-difference-quotient mean of the elements of (multi)set x2 (1-element or 2-element set) under/for function x3.

Let the sumti of x3 be a real-valued, univariate function f which is defined and continuous on at least the closed interval [min(x2), max(x] and differentiable on at least the open interval (min(x2), max(x, such that the derivative f' is injective. Then x1 = (f')^(-1) ((f(max(x2)) - f(min(x if x2 has cardinality 2; if x2 has cardinality 1, then x1 is equal to the sole element of that set; other repairs may be necessary, depending on f: for example, f=log has x1=0 if 0 is an element of x2. If x3 is filled with the monic, single-term polynomial of (integer) degree p such that p>1, then x1 will be the principal (p-1)th root of the (1/p)-scaled weighted (p,1)-Lehmer mean (cnanlime) of the elements of x2 wherein the weight of the lesser of them is -1 (and the other weight is +1) unless x2 is a singleton (in which case the weights are both +1). Usually, x1 is the value which is guaranteed to exist for appropriate function x2 by the Mean-Value Theorem.