User:Bwisialo
The global maximum for the function
occurs at x = e. This functional property of x(1/x) is continuous with the limit property of the function
The two functions are instances of
where x(1/x) occurs at n = 0, and (1+x)(1/x) occurs at n = 1. From n = 0 to n = 1, the minima and maxima of (n + x)(1/x) form a continuous curve from the global maximum of x(1/x) until converging on the limit coordinates of (0, e).
Several properties of exponential functions can be connected with the exponential inequality
where et = (1 + t) only at t = 0.[1] Using this inequality expression, e1/x ≥ 1 + 1/x and, hence, e ≥ (1 + 1/x)x, such that
is an increasing function with a horizontal asymptote at y = e. More generally,
Additionally, the above exponential inequality can be used solve Steiner's Problem.
can be reduced to , yielding the solution that the global maximum for the function
occurs at x = e.
The number e is the unique real number such that
for all positive x.[2] Since it can be proved that
the inequality above provides a demonstration that this limit is e. Also, e is the unique real number a such that
is true for all real x; and e is the unique positive number a such that ax = x + 1 if and only if x = 0, with the result that e is the unique real number a where the slope of ax at x = 0 is equal to 1.
The number e is the unique real number a such that
is true for all real x; and e is the unique positive number a such that ax = x + 1 if and only if x = 0.[3] Also, the inequality ex = 1 + x can be used to prove e is the unique real number such that
for all positive x. Since it can be also be proved that
the inequality expressions above provide a demonstration that this limit is e.
- ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. p. 44-48; 368. Retrieved 13 July 2015.
- ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. p. 44-48.
- ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. p. 44-48.