Let f(x)=a-(x-3)^8 / 9, then greatest value of f(x) is

We have,

$f(x)=a-(x-3)^{8 / 9}$

Clearly, $(x-3)^{8 / 9}>0$ for all $x \in R$

$\Rightarrow f(x) \leq a$ for all $x \in R$.

Hence, the greatest value of f(x) is a.