Section B

Answer this question in the answer booklet provided. Please start each question on a new page.

$7.[$Maximum mark: $14]$

No GDC$.$

Let $f(x)=m{x}^2-2mx,$ where xinR and minR. The line $y=mx-9$ meets the graph of $f$ at exactly one point.

$($a$)$ Show that $m=4.$

$[6]$

The function $f$ can be expressed in the form $f(x)=4(x-p)(x-q),$ where $p,$qinR.

$($b$)$ Find the value of $p$ and the value of $q.$

$[2]$

The function $f$ can also be expressed in the form $f(x)=4(x-h{)}^2+k,$ where $h,$kinR.

$($c$)$ Find the value of $h$ and the value of $k.$

$[3]$

$($d$)$ Hence find the values of $x$ where the graph of $f$ is both negative and increasing.

$[3]$