This question pertains to "A Entropy view of Fibonacci Trees" (1982) by Yasuichi Horibe, Zbl 0491.94009.
Horibe defines a binary tree where a node has a left branch with probability x and a right branch with probability $1-x$. The left branch has cost 1 and the right branch has cost c. In equation (4), p. 171 of the paper, the entropy of a node per cost is provided as $$ { H(x, 1-x)} \over{x + c (1-x)} $$ Horibe states "Let $\lambda$ be the maximizing value of x. By differentiating, $\lambda$ is the unique positive root of $x^c= 1 - x$."
I am afraid I am unable to see how this is found. Can anyone on MO show me? Feeding the question to Mathematica or GPT4 produces no useful answer. Nor do I as yet follow the next sentence "The maximum value of the function is $-\log \lambda$."
Thank you,
Bill