x_{1} is the mean-value theorem mean/forward-difference-quotient mean of the elements of (multi)set x_{2} (1-element or 2-element set) under/for function x_{3}.

Let the sumti of x_{3} be a real-valued, univariate function f which is defined and continuous on at least the closed interval [min(x_{2), max(x}] and differentiable on at least the open interval (min(x_{2), max(x}, such that the derivative f' is injective. Then x_{1 }= (f')^(-1) ((f(max(x_{2)) - f(min(x} if x_{2} has cardinality 2; if x_{2} has cardinality 1, then x_{1} is equal to the sole element of that set; other repairs may be necessary, depending on f: for example, f=log has x_{1}=0 if 0 is an element of x_{2}. If x_{3} is filled with the monic, single-term polynomial of (integer) degree p such that p>1, then x_{1} will be the principal (p-1)th root of the (1/p)-scaled weighted (p,1)-Lehmer mean (cnanlime) of the elements of x_{2} wherein the weight of the lesser of them is -1 (and the other weight is +1) unless x_{2} is a singleton (in which case the weights are both +1). Usually, x_{1} is the value which is guaranteed to exist for appropriate function x_{2} by the Mean-Value Theorem.