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The global maximum of or x(1/x) occurs at x = e. This functional property is continuous with the limit property, limx→0 (1+x)(1/x) = e, as illustrated by the minima and maxima of (n+x)(1/x) for 0 ≤ n ≤ 1.

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

The global maximum of occurs at x = e.

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.

e is the unique positive number a such that axx + 1 holds for all x and such that ax = x + 1 if and only if x = 0.

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.

  1. ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. p. 44-48; 368. Retrieved 13 July 2015.
  2. ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. p. 44-48.
  3. ^ Dorrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover. p. 44-48.