The monotone increasing interval of function f (x) = xlnx on (0,5) is______ ．

The monotone increasing interval of function f (x) = xlnx on (0,5) is______ ．

∵ f ′ (x) = LNX + 1, Let f ′ (x) > 0: LNX ＞ - 1, ∵ x ＞ E-1 = 1E. The monotone increasing interval of ∵ function f (x) = xlnx on ∵ x ∈ (0, 5) is (1E, 5). So the answer is: (1E, 5)

Using the mean value theorem to prove inequality of higher numbers
1-1/x＜lnx＜x-1 （1＜x）
How to solve it

Certification:
Let LNF (x) = 1
LNX = lnx-ln1 = f '(1 + θ x) (x-1) = (x-1) / (1 + θ x), θ∈ (0,1)... Lagrange mean value theorem
∴1+θx∈(1,1+x)
∴1-1/x

Mean value theorem should be used to prove inequality
x/(1+x)0)
|arctan a-arctan b|

(1) If f (x) = ln (1 + x) is defined without looking at the domain x = 0, then f (x) is continuous in the interval [0, x] and there must be a point that satisfies ln (1 + x) - ln (0 + 1) = 1 / (1 + &) * (x-0) where 01 obviously in (1 + x) B removes the absolute values on both sides (a = B is better proof) first function f (x) = arctanx