Let $a$ and $b$ be real numbers. Consider the cubic equation $$x ^ { 3 } + 2 b x ^ { 2 } - a ^ { 2 } x - b ^ { 2 } = 0$$ (i) Show that if $x = 1$ is a solution of ( $*$ ) then $$1 - \sqrt { 2 } \leqslant b \leqslant 1 + \sqrt { 2 }$$ (ii) Show that there is no value of $b$ for which $x = 1$ is a repeated root of ( $*$ ). (iii) Given that $x = 1$ is a solution, find the value of $b$ for which $( * )$ has a repeated root. For this value of $b$, does the cubic $$y = x ^ { 3 } + 2 b x ^ { 2 } - a ^ { 2 } x - b ^ { 2 }$$ have a maximum or minimum at its repeated root?
(i) [5 marks] As $x = 1$ is a root then
\section*{2. For ALL APPLICANTS.}
Let $a$ and $b$ be real numbers. Consider the cubic equation
$$x ^ { 3 } + 2 b x ^ { 2 } - a ^ { 2 } x - b ^ { 2 } = 0$$
(i) Show that if $x = 1$ is a solution of ( $*$ ) then
$$1 - \sqrt { 2 } \leqslant b \leqslant 1 + \sqrt { 2 }$$
(ii) Show that there is no value of $b$ for which $x = 1$ is a repeated root of ( $*$ ).\\
(iii) Given that $x = 1$ is a solution, find the value of $b$ for which $( * )$ has a repeated root. For this value of $b$, does the cubic
$$y = x ^ { 3 } + 2 b x ^ { 2 } - a ^ { 2 } x - b ^ { 2 }$$
have a maximum or minimum at its repeated root?