Answer the following questions (a) A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_{1}, \ldots, a_{k}$, each $a_{i} > 1$, such that $$\frac{1}{a_{1}} + \cdots + \frac{1}{a_{k}} = 1.$$ Show that if $k$ is stable then $k+1$ is also stable. Using this or otherwise, find all stable numbers. (b) Let $f$ be a differentiable function defined on a subset $A$ of the real numbers. Define $$f^{*}(y) := \max_{x \in A}\{yx - f(x)\}$$ whenever the above maximum is finite. For the function $f(x) = -\ln(x)$, determine the set of points for which $f^{*}$ is defined and find an expression for $f^{*}(y)$ involving only $y$ and constants.
Answer the following questions
(a) A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_{1}, \ldots, a_{k}$, each $a_{i} > 1$, such that
$$\frac{1}{a_{1}} + \cdots + \frac{1}{a_{k}} = 1.$$
Show that if $k$ is stable then $k+1$ is also stable. Using this or otherwise, find all stable numbers.
(b) Let $f$ be a differentiable function defined on a subset $A$ of the real numbers. Define
$$f^{*}(y) := \max_{x \in A}\{yx - f(x)\}$$
whenever the above maximum is finite.
For the function $f(x) = -\ln(x)$, determine the set of points for which $f^{*}$ is defined and find an expression for $f^{*}(y)$ involving only $y$ and constants.