Determine all possible positive real numbers X that satisfy this equation:

X^0.4X = 0.4^X0.4

Note: For the purposes of the problem, a^bc is equal to a^(b^c)

(In reply to computer exploration (spoiler) by Charlie)

Seeking to find the exact point where the LHS reaches its maximum, I found it easier to differentiate its logarithm, which is y = ln(x)*0.4^x.

y'=.4^x *(1/x + ln(0.4)*ln(x))

This could be solved iteratively through:

DEFDBL A-Z
x = 1.1
DO
  x = EXP(1 / (x * LOG(5 / 2)))
  PRINT x, x ^ (.4 ^ x)
LOOP

which finds the LHS equal to approximate 1.119625892520564 when x is approximately 1.820921921969026.

Posted by Charlie on 2008-03-10 17:42:56