2024 NJC Promo Q8

The diagram below shows that the graph of \(y = \mathrm{g}'(x)\) has two \(x\)-intercepts at \((-1,0)\) and \((1,0)\) as well as two stationary points whose \(x\)-coordinates are \(-1\) and \(\frac{1}{2}\) respectively.

1. A stationary point on the curve of \(y = \mathrm{g}(x)\) has a \(x\)-coordinate that is positive. State the value of this \(x\)-coordinate and determine the nature of this stationary point on the curve \(y = \mathrm{g}(x)\).[2]

It is given that \(\mathrm{g}'(x) = ax^4 + bx^3 + cx^2 + dx + e\).

2. Show that \(a - b + c - d + e = 0\).[1]

Given further that the gradient of the tangent at the point of inflexion on \(y = \mathrm{g}(x)\) is \(-\frac{27}{8}\), find the values of \(a\), \(b\), \(c\), \(d\) and \(e\).[4]