, 2022

This question requires answers only.
Define $f(x)$ to be the integer part of $\sqrt[3]{x}$; for example $\sqrt[3]{3.375} = 1.5$ so $f(3.375) = 1$, $\sqrt[3]{9} \approx 2.08$ so $f(9) = 2$, and $\sqrt[3]{27} = 3$ so $f(27) = 3$.
(a) Write down the first six positive cube numbers. Hence write down the value of $f(122)$. (1 mark)
(b) If $x$ is a positive integer with $f(x) = 3$, find the possible values of $f(2x)$. (3 marks)
(c) Find all positive integer values of $x$ such that $f(x) + f(2x) + f(3x) = 10$. (6 marks)