MathDB

Answer the following questions :

<span class='latex-bold'>(a)</span>~

A natural number

k

is called stable if there exist

k

distinct natural numbers

a_1, a_2,\cdots, a_k

, each

a_i>1

, such that

\frac{1}{a_1}+\frac{1}{a_2}+\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.

<span class='latex-bold'>(b)</span>

Let

f

be a differentiable function defined on a subset

A

of the real numbers. Define

f^*(y):=\max_{x\in A} \left\{yx-f(x)\right\}

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.

1