Daily questions _190915

About known \ (X \) equation \ (x ^ 2 \ ln x = a \ ln aa \ ln x \) has \ (3 \) distinct real roots, find \ (A \) in the range of .
Analysis: i.e. on the original title \ (X \) equation \ (\ ln x- \ dfrac { a \ ln a} {x ^ 2 + a} = 0 \) has three distinct real roots. Note
\ [f (x) = \
ln x- \ dfrac {a \ ln a} {x ^ 2 + a}, x> 0, a> 0. \] to \ (f (x) \) derivative can be have \ [f '(x) = \ dfrac 1x + \ dfrac {2a {\ ln a} x} {(x ^ 2 + a) ^ 2} = \ dfrac {x ^ 4 + 2a (1+ \ ln a) x ^ 2 + a ^ 2}
{x (x ^ 2 + a) ^ 2}. \] to \ (f (x) \) there are three different zero, then \ (f '(x) \ ) at least there are two variations number zero, i.e. the equation \ [x ^ 4 + 2a (
1+ \ ln a) x ^ 2 + a ^ 2 = 0 \] there are at least two positive roots, so
\ [\ begin {cases} & \ Delta = 4a ^ 2 (1+
\ ln a) ^ 2-4a ^ 2> 0, \\ & -2a (1+ \ ln a)> 0, \ end {cases} \] solution to give\ (0 <A <\ dfrac. 1 {\ RM ^ 2} E \) . At this time, \ (f '(x) \ ) in \ ((0, + \ infty ) \) have a different and only two zeros, to \ (m, n-\) , and \ (m <n \) . Thus \ (f (x) \) in \ ((0, m), (n, + \ infty) \) monotonically increasing, the \ ([m, n] \ ) monotonically decreasing. Noting \ ( F \ left (\ sqrt {A} \ right) = 0 \) . and
\ [f '\ left (\ sqrt {a} \ right) = 2a ^ 2 \ left (2 + {\ ln} a \ right) <0. \]
so \ (0 <m <\ sqrt {A} <n-\) , so
\ [f (m)> f \ left (\ sqrt {a} \ right) = 0> f (n). \]
take \ (x_2 = n-+. 1> n-\) , then
\ [\ forall a \ in \ left (0, \ dfrac {1} {\ mathrm {e} ^ 2} \ right), f (x_2) > 0- \ dfrac {a {\
ln} a} {x ^ 2 + a}> 0. \] take \ (x_1 = \ dfrac mA} {A} + {m <m \) , then
\ [\ Forall a \ in \ left (0, \ dfrac {1} {\ mathrm {e} ^ 2} \ right), f (x_1) <{\ ln} a- \ dfrac {a {\ ln} a } {0 2 + a} =
0. \] ^ summary, when \ (0 <a <\ dfrac {1} {\ mathrm {e} ^ 2} \) , the function \ (f (x) \) in the three sections of monotonic sections respectively a zero. and three zeros are located in the interval
\ [(x_1, m),
(m, n), (n, x_2). \] Thus ask \ (a \) taken value range \ (\ left (0, \ dfrac. 1} {{{\ RM E ^ 2}} \ right) \) .