1.32: Past-paper question 7

Two numbers $\displaystyle x$ and $\displaystyle y$ are such that $\displaystyle 3x - y = 4$

(a) Show that $\displaystyle S = 5x^3 + 9x^2 - 24x + 16$

Given that $\displaystyle x$ can vary,

(b) use calculus to find the value of $\displaystyle x$ for which $\displaystyle S$ is a minimum, justifying that this value of $\displaystyle x$ gives a minimum value of $\displaystyle S$

(c) Find the minimum value of $\displaystyle S$