Express \(\cos 4 \theta\) in terms of \(\sin 2 \theta\) and hence show that \(\cos 4 \theta\) can be expressed in the form \(1 - k \sin ^ { 2 } \theta \cos ^ { 2 } \theta\), where \(k\) is a constant to be determined.
By expressing \(2 \cos ^ { 2 } 2 \theta - \frac { 8 } { 3 } \sin ^ { 2 } \theta \cos ^ { 2 } \theta\) in terms of \(\cos 4 \theta\), find the greatest and least possible values of
$$2 \cos ^ { 2 } 2 \theta - \frac { 8 } { 3 } \sin ^ { 2 } \theta \cos ^ { 2 } \theta$$
as \(\theta\) varies.
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The function f is defined for all real values of \(x\) by
$$\mathrm { f } ( x ) = k \left( x ^ { 2 } + 4 x \right) ,$$
where \(k\) is a positive constant. The diagram shows the curve with equation \(y = \mathrm { f } ( x )\).
The curve \(y = x ^ { 2 }\) can be transformed to the curve \(y = \mathrm { f } ( x )\) by the following sequence of transformations: a translation parallel to the \(x\)-axis,
a translation parallel to the \(y\)-axis,
a stretch. a translation parallel to the \(x\)-axis, a translation parallel to the \(y\)-axis, a stretch.
Give details, in terms of \(k\) where appropriate, of these transformations.
Find the range of f in terms of \(k\).
It is given that there are three distinct values of \(x\) which satisfy the equation \(| \mathrm { f } ( x ) | = 20\). Find the value of \(k\) and determine exactly the three values of \(x\) which satisfy the equation in this case.