A curve, \(C\), has equation \(y=(2x+8)^{\frac32}+3x^2\), where \(x>-4\).
Find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\).[1]
Duffy claims that two of the tangents to \(C\) are parallel to the straight line \(6x+y+4=0\). Determine if Duffy is correct. Show your working clearly.[6]
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Answer:(a) \(3\sqrt{2x+8}+6x\). (b) Incorrect; only one such tangent, at \(x=-2\).